Hamiltonian Picture of Light Optics First-Order Ray Optics

7/30/2019 Hamiltonian Picture of Light Optics First-Order Ray Optics

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1Ham i l ton ian P ic tu r e o f L igh tF i rs t -Order Ray Opt ics Opt ics .

1.1 In t roduct ionT h e p h a s e s p a c e r e p r e s e n t a t i o n o f l ig h t o p t ic s n a t u r a l l y a r i se s f ro m t h e H a m i l -t o n i a n f o r m u l a t i o n o f g e o m e t r i c a l o p t i c s. G e o m e t r i c a l o p t i c s g iv e s a, s i m p l em o de l fo r l i gh t behav i ou r , i n wh i ch t he wave cha rac t e r o f l i gh t is i gno red . I t isv a l i d w h e n e v e r l i g h t w a v e s p r o p a g a t e t h r o u g h o r a r o u n d o b j e c t s w h i c h a r e v e r yl a rg e c o m p a, r e d t o t h e w a v e l e n g t h o f t h e l ig h t a n d w h e n w e d o n o t e x a m i n e t o oc lo s e ly w h a t is h a p p e n i n g i n th e p r o x i m i t y o f s h a d o w s o r f oc i. A c c o r d i n g l y , i tdoes no t accoun t fo r d i f f r ac t i on , i n t e r f e rence o r po l a r i za t i on e f f ec t s . Geom et -r i c a l o p t i c s e m p l o y s t h e c o n c e p t o f light ray [1] , wh i ch we m ay g i ve t he na i vev i ew as a,n i n f i n i t e s i m a l l y t h i n be am o f l i gh t. Seve ra l fo rm a l de f i n i t ions o f li gh tr a y h a v e b e e n e l a b o r a t e d w i t h i n b o t h t h e c o r p u s c u l a r a n d w a v e t h e o r y t o a c -c o m m o d a t e g e o m e t r i c a l a b s t r a c t i o n a n d p h y s i c a l o b s e r v a b i l i t y . A l l d e f i n i t i o n sw o r k w e l l i n c e r t a i n s i t u a t i o n s , b u t i n o t h e r s a r e c o n f r o n t e d w i t h i n t r i n s i c a l l yphys i ca l d if f icu l ti e s . Thus , fo r i n s t anc e , t he c o rp usc u l a r v i ew o f r ays a s t r a -j e c t o r ie s o f " l u m i n o u s " c o r p u s c le s c o n f r o n t s w i t h t h e p r o b l e m t h a t t h e e n e r g yd e n s i t y m a y b e c o m e i n f i ni te . L i k e w is e t h e w a v e - l ik e v ie w o f r a y s a s o r t h o g o n a lt r a j e c t o r i e s t o t h e p h a s e f r o n t s o f t h e l ig h t w a v e c o n f r o n t s w i t h t h e d i f f ic u l tyo f i n d i v i d u a l i z i n g a d e f in e d w a v e f r o n t i n t h e t w o - w a v e o v e r l a p d i s t r i b u t i o n .I n d e e d , t h e r a y m u s t b e t h o u g h t o f a s a c o n v e n i e n t a n d s u c c e ss f u l m o d e l w h i c hs u p p o r t s o u r p e r c e p t i o n , a n d h e n c e f a c i l it a t e s t h e f o r m a l d e s c r i p t io n , o f a w i d ec l as s o f l ig h t p h e n o m e n a . G e o m e t r i c a l o p t i c s e s t a b l is h e s t h e g e o m e t r i c a l r u le sg o v e r n i n g t h e p r o p a g a t i o n o f l ig h t ra y s t h r o u g h o p t i c a l s y s t e m s .

T h e a n a l o g y o f g e o m e t r i c a l o p t i c s o f l ig h t r a y s t o H a m i l t o n i a n m e c h a n i c so f m a t e r i a l p a r t i c l e s i s w e ll e s t a b l i s h e d a n d e f fe c ti v e ly e x p l o i te d . T h e Hamil-tonian formal ism was o r i g i na l l y deve l oped by Ham i l t on fo r op t i c s i n h i s 1828p a p e r Theory of Sys tems of Rays a n d i n s u b s e q u e n t p a p e r s a n d b r ie f n o t e s,pub l i shed du r i ng t he ye a r s f rom 1830 t o 1837 [2 .1 ]. In h i s pap e r s , H am i l t o n


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2 L i n e ar R a y an d W a v e O p t ic s n P h a s e S p a c e

f o r m u l a t e s t h e p r o b l e m o f s t u d y i n g t h e g e o m e t r y o f l ig h t ra y s a s t h e y p a s st h r o u g h o p t i c a l s y s t e m s i n t e r m s o f w e l l - d ef i n e d r e l a t i o n s b e t w e e n t h e l oc a lc o o r d i n a t e s o f t h e r a y s e n t e r i n g a n d e m e r g i n g f r o m t h e s y s t e m , s p e c if i e d w i t hr e s p e c t t o t h e o p t i c a l a x i s a n d p r o t ) e r l y c h o s e n p l a n e s a c r o s s t i l e a x i s . H es h o w s t h a t , i f t h e r a y c o o r d in a . t e s a, c s u i t a b l y d e f in e d , t h e i n p u t - o u t p u t r c l a;t ions con f igu re a , s symp lec t i c t ra , n s f o r n m t io n s , g e n e r a t e d b y a, f u n c t i o n o f t h eray var ia , t ) lcs , the charact( '~ri ,~tic f u nc tio n, w h o s e f lm ( : t i o n a . 1 f o r m i s d e t e r m in e ds()lely | )y tl le ()t) t i( 'al I )r ot) ert ies ()f t, tw, syste~ll . Im.t( ;r , t l m lfi lt( m realiz(~(1 tha.tth e Sa,lll(; nlet, h()(t ( :olfl(t t)(~ a,I)t)lie(1 lm t:h ang e( 1 t() nl(;( ' lm,lfi(:a,1 sy st e m s, rct)la(: -ing th e o t ) t i ( :a l ax is t )y the ti~ llC a ,x is, the l igh t ray s 1 )y t im t )ar t i ( :h ; t ra j ( ; ( : to r iesa n d th e ray -( :( )( )r( lin atc s t)y l, ll(; lll(;(:lm,ni(:a,1 I)lm,s(;-sI)a(:(; va,ria,lfics [2.2]o

T h e t) ha se st)a(:(; rct)r( '~s(;ld, a.ti()ll is a, fa.llfilia,r nlct, ll()(t wil, llill tll(; H all filt (m ia, nfo rn m la ti o n ()f ( ' lassi( ' ,a,1 nwclm~ fi(:s, wlfi(:ll dcs cril )( ;s t, tlc (lylm ,llfics ()f a. Ill(;(:lm,n-i(:a.1 syst, (; nl w it ll 'll~ (l(~.grt',cs ()f fl'(',c(l()lll ill t(;rll lS ()f '/l~ g('.l~('.ralizc(t ill( l(; I)e nttt mt('()( )r(tinat( ' ,s (q~, q~, ..., (1,,,) aal(l t,l~c sa~ (' , ~ f i ) ( ' . r ()f ( 'aI~()~ i(:a,lly ( ' ()~.j~ga,t( ', va,ri-al) lcs (p, ,p~ , ... ,p, ,,) [3]. TI~(: l~(;(:l~a,l~i(:al l)lm,s(; st)a(:c is t,l~(; C art( ;si m~ st)a.(:c oftl~(;se 2,ttt, (:()()l'(li~m.t(,.s. F () r ('Xa,liq)lc., ill(; st, at( ; ()f a fl'(' ,e t)a, rti(: h' a t a, (: crta .int, n~c is r(~.t)r( '.s(;nt( ' .(t in tim. i)r( )t) cr 6 D t) ha sc st)a,(:(; 1)y a 't'(:pr(:,s('.'nlativ(: p oint ,st)e(:if ic(t l)y t ,h( '. C a r t( ' .s ia~ ( ' ( )()r (li~mt(;s q = (q. ,, , % , q:) m~(l t l~( ' , r ( ; l( ;van t m()-~(',l~l,a. p = (p:,,, p,/, p: ) . ~l' l~(' ~() l,i(n~ ()f tl~(' , t)a.rt, (:l(; i~ r(;al st)a.(:e (:()rl ' (;st)()n(ist() a, t, ra,j(~.(:t()ry in 1)lm,se sI)a(:(;. T h e n , l,l~(~, st, a.t,('. ()f a n (ms(',~fl)l('. ()f i(l(;~d,i(:al an ( tn ( )n in t , e r a ( : t i~ g t)a,rti(:l(' ,s a,l, a gi vc ~ t, ilil(', ( : ( ) r I ' ( ' , s I ) ( ) l l ( t s t ( ) a , s ( ; l , ( ) f I)()iI~t,s i~ t , l~(; 6 Dt)ha s( ' s t)a( ' ( : . Tl~( ' ( l( )~ min ()( '( '~q)i( ;(l t)y this set ()f t)()i~t,s ~( )v( :s t . lm )~ gh i)has(;sI)a,(:e a.s th(', I)a.rti(:l(',s ~ () v('. ii~ i'(' ,al sl)a,(:c, tt () w( we r, a,s t,l~(', t()ta,l ~ n~ fi) cr ()ft)a,r ti( : le.s r( ', l~m,i~ ( :() nsta,nt, s() will th e t() tal mm fi)( ;r ()f I )has(; sI)a,( :(; I )()ints. E vi-(l en tly a, rca,1 (t( ; ~sity ca n t)( ; ass()( : ia.t(;(1 w it h th e r ( ; t ) I ' ( ' , S ( ; l l t a , I , i v ( ; l ) ( ) i l i ts in phasest)a,( :c, a,n(t ( :()rre, sI)( )~ (tingly a. ( t istr it)~d,i()I~ f lln(:t i( m ()f ( l( ;~ sity p( q, p, t) ( :a.n be(lefin(.~(1 s o tha,t p ( q , p , t ) d V st)e( ' if ies the m~nfl)cr of rct)r ( ' .S(;ld;ativ(; t)() ints in thee l e m e n t o f v ( ) l uI n e d V in t tw , v ic i n i ty ( ) f the t ) ( ) in t (q , p ) . L i ( )uv i l le ' s th eo re msta , tes the. i~w~,r ian( ' e of the ( le nsi ty of re t ) res entat i v ( ' , t ) () i~d,s a ,l ( )~g I, m tra je c-to r y o f any t ) ( ) in t , an d a ( : ( ' o r ( t ing ly ( ) f th( '. vo lm ne o f the I )hasc spa,( '.e do ma in ,even th ou gh i t s sha .pc ma y ( : l~a,ng (; ( :ons i ( t ( ; r ab ly ( lu r ing th e m o t i on .

L ik e w i s e t h e g e o m e t ri ( : -o t ) ti c a ,1 t ) h a, se s t ) a c e i s t h e 4 D C a r t e s i a n s p a c e o ft h e r a y p o s it i on a n d m o m e n t u m c o o r d i na t e s (q z , qy , p z , p y ) . H o w e v e r , t il e p h a s es t ) ac e s of c l a s si c a l m e c h a n i c s a n d g e o m e t r i c a l o p t i c s a r e g l o b a l ly d i f fe r e nt . T h et)a, r t i c le m o m e n t u m o f c l a s s ic a l m e c h a n i c s is n o t r e s t r i c t e d i n v a l ue , w h i l s t t h er a y m o m e n t u m o f g e o m e t r i c a l o p t ic s i s c o n fi n e d w i t h i n a c i rc l e d e t e r m i n e d b yt h e l o c a l r e fr a ,c t iv e i n d e x t h r o u g h t h e i n h e r e n t f o r m o f t h e o p t i c a l H a m i l t o n i a n .I n t h e l in ( ;a r a p p r o x i m a t i o n t h e r a y m o m e n t u m is a s s u m e d t o r a n g e w (; ll b e l o wi t s n a , t u r a l l im i t , w h i c h t h e n i s i g n o r e d . T h u s t h e g e o m e t r i c - o p t i c a l p h a s e s p a c eo f l in e a r o p t i c s c o m e s t o b e s i m i l a r to t h e m e c h a n i c a l p h a s e s p a c e .

S e c t i o n 1 .2 r e v i e w s t h e H a m i l t o n i a n f o r m u l a t i o n o f g e o m e t r i c a l o p t i c s a n d


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Ham iltonian Picture of Light Optics. First-Order Ray Optics 3

Lagrangian picture Solves Lagrange ' s equat ions for the Particle trajectoryin real spaceLegendreHam ilton's pr inciple transformation

Hamiltonian picture Solves Hamil ton ' s equat ions for the Particle trajectory9 . - in phase space

F e r m a t ' s principleLagrangian picture Solves Lagrange ' s equat ions for the9 9

LegendretransformationHamiltonian picture S o lve s Ha mi lum ' s e qua t io n s fo r th ~

Ray trajectoryin real space

Ray trajectoryin phase space

F I G U R E 1 .1 . T h e F e r m a t e x t r e m a l p r i n c i p l e b a s e d f o r m u l a t i o n o f g e o m e t r i c a l o p ti c s m i r ro r st h a t o f c l as s ic a l m e c h a n i c s , b a s e d o n t h e H a m i l t o n m i n i m a l p r i n ci p l e .

i n t r o d u c e s t h e r e l a t e d c o n c e p t o f g e o m e t r i c - o p t i c a l p h a s e s p a c e . S e c t i o n 1 .3e m p h a s i z e s t h e s y m p l e c t ic n a t u r e o f r a y p r o p a g a t i o n , a n d d e t a il s t h e s u i t e dm a t h e m a t i c a l s e t t in g s ( P o is s on b r a c k e t s a n d L i e o p e r a t o r s ) t o a p p r o a c h t h e i n-t e g r a t i o n o f H a m i l t o n ' s e q u a t i o n s f o r t h e l ig h t r ay . In S e c t. 1 .4 t h e r a y - t r a n s f e ro p e r a t o r i s i n t r o d u c e d a n d t h e r e le v a n t L i e - t r a n s f o r m a t i o n b a s e d f o r m a l i s m isd e s cr ib e ( t. I l l u s t r a t i v e e x a m p l e s o f p h a s e - st ) a c e t r a n s f o r m a t i o n s a r e g i v en i nSec t . 1 . 5 . Sec t ions 1 . 6 and 1 . 7 i l l us t r a t e t he l i nea r appr ox ima t ion to l i gh t - r ayp r o p a g a t i o n , w h i c h n a t u r a l l y y i e l d s t h e r a y - t r a n s f e r m a t r i x f o r m a l i s m . F i n a l l y ,S e c t . 1 . 8 c l a r i f i e s t h e l i n k b e t w e e n t h e r a y - m a t r i x a p p r o a c h a n d t h e c a r d i n a lp o i n t ( a n d p l a n e s ) m e t h o d .

1.2 Ha m iltonian picture of light-ray propagationW e w i ll g i ve a b r i e f a c c o u n t o f t h e H a m i l t o n i a n f o r m u l a t i o n o f g e o m e t r i c a lo p t i c s i n o r d e r t o f i x t h e n o t a t i o n s w e a d o p t a n d t o t r a c e t h e c o n c e p t u a l p a t ht o w a r d s t h e I ) h a s e s p a c e r e p r e s e n t a t i o n a n d t h e i n h e r e n t g e o m e t r y .

H a m i l t o n i a n o p t i c s d e v e l o p s f r o m F e r m a t ' s p r i n c i p l e o f e x t r e m a l o p t i c a lp a t h , w h i c h i s t h e o p t i c a l a n a l o g o f H a m i l t o n ' s p r i n c ip l e o f l e a s t a c t io n ( F i g .1 .1 ). F r o m H a m i l t o n ' s p r i n ci p le o n e c a n d e r iv e b o t h t h e L a g r a n g i a n a n d H a m i l -t o n i a n m e c h a n i c s , r e la t e d t h r o u g h t h e L e g e n d r e t r a n s f o r m a t i o n [3 ]. L i ke w is ef r o m F e r m a t ' s p r i n c i p l e o n e c a n d e v e l o p t h e L a g r a n g i a n a s w e l l a s t h e H a m i l -t o n i a n f o r m u l a t i o n o f o p t ic s [4 ]. T h e f o r m e r y i e ld s t h e e q u a t i o n s f or t h e r a yv a r i a b l e s i n r e a l s p a c e , w h i l e t h e l a t t e r t h e e q u a t i o n s f o r t h e r a y v a r i a b l e si n p h a s e s p a c e . W e w i l l c u r s o r i l y i l l u s t r a t e t h e b a s i c s t e p s l e a d i n g t o b o t h


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4 L i n e a r R a y an d W a v e O p t i c s n P h a s e S p a c e

n ( x , y , : )

x \ s

r (s ) = ( x , y , : )

Z

F IG U R E 1.2. Ge()Iimtri('al ()I)tics (h;scril)cs t, le Ine(li~In by tt,e refl'm'tive i,l(lex flu,ction'n(x, y, z) m~ (l t,l,e ligld, r ays 1)y l, ie 3-vc(:t()r ()f f,l,,(:ti(),~s r( s) = (:r(.~) , y(.s), z(.s)) ()f th e arcle~gt| ~ .s Ineas~m;(l alo ~ g l,|~e ray I)al,]~.

t)i(:t, lu' es, a,( t( h'e ssi lig t, i(' I'(~a,(h;r t() [4] fl)r a lii() I 'e (t( ~ta ih;(t t,r(; atili( ',lit.As a ,m t l m d s( : (~iiar i( ) f ( )r i l i tr ( ) ( l l i ( : i l lg F ( ;n lm t ' s t )r i l l ( : i t )h ; [4, 5], w(; ( : ( ) l ls i( ter

a,n inh( )nl () g(m (;( )lts 111(;(lilnll, ()(: (: lH) yillg a, (: (' ,r l,a,ill re gi( )n ill 1,11(; 3 D Sl)a(:e , w h er ew(; sllt)l)()s(; a (~,a.rl,(;sia,ll sysI,(; lli ( )f (: ()() r(lilml,(;s (x , y, z) |)(; a ssig lle (l . 'I'11(; ()t)ti-(:a,1 i)i ' ()t)ert, i(;s ()f ti m lll( ' ,til ull a,r(; ty I)i( 'a,lly (t(;s(:i ' il)e(1 | )y t,ll( ' r( ',f l'a,(: tiv(; in( te x'n,(:r , y, z) , giv( '~Ii as a. s(~ala,r fllll(: t,i()II ()f sire(:(; ( ' ' ) . A li gh t ra y is I)r( )t) aga,t ing inl,h(; in( ;( lil nn a,h) ng s()~ll( ' , I ,raj(; (: t( )ry. R,( ;gar( h' ,l a,s a, lin e in l, ll(; 3 D st)a,(: (;, th e ra y(:a,,, a,(:( :oi'( tiIlgly t)e (h;s('rit)( ;(1 l)y t, tl(; l)()sit, i()11 ve(;t~()r r ( s ) ---- ( . r ( . s ) , y ( , s ) , z ( s ) )for t )oint~s ()n 1,11(; ray, w it h th( ; ( : ( ) ( )r ( l ina,t, ; s t ) e in g f lm(: t , i ( )ns ()f l, l ( ; a ,I '( : le ng th,l l ( ;a ,s lu 'e( t a l( ) l lg 1,11(; ra y t)a , t ll wi tl l res t)e( : t , t ( ) a ( :h osen t) ( ) ild , (F ig. 1 .2) .

F er nm , t, 's t) I ' ill(: it)h; (: ()1111)ill(;s 1,11(; ge()nlet~ri(:a,1 an(1 t) hys i( :al a,sl)(; (: ts ()f th e ra yt)rot)a, ga ti ol l t, lr( )ll gh the, ( ' ()n(:( ',i)t, ()f o p t i c a l p a t h .

W e r e c a l l th a t g iv e , l tw( ) l ) ( ) i ld , s P , a ,n (t P ~ a n ( t a ( : l lr v e C ( ' ( ) ,l n ( ;( : t il l g th e m,t h e g e o m e t r i c a l t )a ,t , l leng t t l 12 (C ) f rom P~ t , ( ) P2 a l on g C is ( le t il le(1 as t hele n g th o f C a ,n (t he n (: ( ; is f ( )n n a , l ly g iv e n b y th e l in e in te g r M

P 211 (C ) -- d,s, (1 .2. 1)1

t ) e r f o r m e d a l o n g C f r o m P1 tO P 2 ; s d e n o t e s t i le a r c l e n g t h m e a s u r e d a l o n g t h ep a t h a n d d s = v / a x 2 + ay 2+ d z 2 i s t h e i n f i n i t e s i m a l a r c l e n g t h .

C o r r e s p o n d i n g l y , t h e o p t i c a l p a t h l e n g t h s ( C ) a l o n g t h e r a y t r a j e c t o r y Ca W e w i l l c o n s i d e r o n l y l i n e a r s p a t i a l l y n o n d i s p e r s i v e i s o t r o p i c m e d i a , w h o s e r e f r a c t i v e

i n d e x f l ln c t io n i s a c c o rd i n g l y d e p e n d e n t o n p o s i t i o n a n d i n d e p e n d e n t o f d i r ec t io n . H e n c e w ew i l l d i s t in g u i s h o n l y b e t w e e n h o m o g e n e o u s a n d i n h o m o g e n e o u s m e d i a , a c c o r d i n g t o w h e t h e rt h e s c a la r i n d e x f u n c t i o n i s u n i f o r m o r ch a n g e s f r o m p o i n t t o p o i n t w i t h i n t h e m e d i u m .


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Ham iltonian P icture o f Light Optics. First-Order Ray Optics 5f rom P1 to P2 is defined as the l ine in teg ra l a long C of the ref rac t iv e index"

s ( C ) - n ( x , y , z ) d s . (1 .2.2)1

I f t h e m e d i u m is h o m o g e n e o u s , s o t h a t n ( x , y, z ) - n o , t h e o p t i c a l p a t h l e n g t his t h e g e o m e t r i c a l p a t h l e n g t h m u l t i p li e d b y t h e r ef r a c ti v e in d e x: s - n 0 s

Sl igh t ly co r r ec t ing t i l e o rig inM fo rmu la t ion , a s g iven by F e rm at in h i s(Evreus (1891) [5.1]" " J e r e c o n no i s p r e m i ~ r e m e n t . . . l a v d r i td d e c e p r i n c i p e ,q u e l a n a t u r e a g i t t o u j o u r s p a r l e s v o i e s l e s p l u s c o u r t e s , " F e r m a t ' s p r i n c i p l es t a t e s t h a t , a m o n g a ll t h e p o s s ib l e p a t h s C c o n n e c t i n g t h e p o i n t s P 1 a n d P 2 ,the l igh t r ay w ou ld fo llow the pa th C whos e op t i ca l pa th l eng th /2 (C ) i s ane x t r e m u m . T h e r e f o r e i t m a y b e a m i n i m u m , w h i c h i s t h e m o s t f r e q u e n t c a s e ,a m a x i m u m , o r i t m a y b e s t a t i o n a r y w i t h r e s p e c t t o t h e o p t i c a l p a t h l e n g t h so f o t h e r p a t h s c l o se ly a d j a c e n t t o C . I n m a t h e m a t i c a l t e r m s , t h e a c t u a l r a ypa th C i s iden ti f ied a s the ex t r e m al o f the va r i a t iona l p rob lem

~;z; (C) - ~ n ( x , y , z ) d s - O , (1 .2 .3 )1

wh ere the ~ va r i a t ion is in t ended fo r s ma l l dev ia t ion s w i th r e s pe c t to C o f thei n t e g r a t i o n p a t h b e t w e e n t h e t w o f i x e d e n d p o i n t s P 1 a n d P 2 .

T h e f o rm a l c o r r e s p o n d e n c e o f F e r m a t ' s p r i n c i p le ( 1 .2 .3 ) w i t h H a m i l t o n ' sp r i n c i p l e b e c o m e s a p p a r e n t , o n c e c h a n g i n g t h e i n t e g r a t i o n v a r i a b l e f r o m sto one of the Car tes ian coor ( t ina tes , say z . Then , ev i ( lenci l lg the ( t i f feren t ia le l e m e n t d z in the l ine e lement d s , w e m a y c a s t F e r m a t ' s p r i n c i p l e ( 1 . 2 . 3 )exac t ly in the s ame fo rm as Hami l ton ' s p r inc ip le , i . e . ,

(~ {~(X, y , Z )v / l+x '2+y '2}dz=O , (1.2.4)1

where the in teg rand func t ion i s na tu ra l ly iden t i f i ed a s the o p t i c a l L a g r a n g i a nL ( x , y , x ' y ' z ) - n ( x y z ) v / l + x '2 + y '2

w i t h(1.2.5)

x~ d x y~ dyd z ' d--~ " (1 .2 .6 )T h e L a g r a n g i a n s y s t e m o f e q u a t i o n s p l a i n ly f o llo w a s

d (OL_=__:.) O L O,d z ~ " O ~d O L O L(~ y--u ) O y o ,d z

(1 .2.7)


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6 L i n e a r R a y a n d W a v e O p t i c s n P h a s e S p a c e

a n d , o n a c c o u n t o f ( 1 .2 .5 ) a n d ( 1 .2 . 6 ) , c a n b e r e a r r a n g e d i n t o a, s i n g l e v e c t o r i a ld i f f er e n t ia l e q u a t i o n o f s e c o n d o r d e r , k n o w n a s t h e r a y e q u a t i o n ,

d ( n d rds ( -~s ) - V n , (1 .2 .8 )re l a t ing the ray t ra , j cc to ry t () t he o t) ti ca ,1 t ) ro t ) e r t i e s ( )f t he ~ne ( t ilml [6 ] . Inde ed ,_d~ is t h e ta n g ( u lt t(~ t h e ra,y t)a.tl~, a,n(1 V n - (~ ~ o,,,,o:,j o,,,,o~) h~ t( ;r es tin gl y, E q.d,N ~ ~ " .(1.2 .8) sh ow s a noti (:oat) l( '~ resel lfl ) lal l ( :e t ( ) th e e( t lm.t i(m ()f ( : lassi( :al rclat ivist i ( ' ,nle(:ha,Ili(:s F - ~ tl m t I ' llh',s th e (lyn anf i(: s ()f a I)(fillt-t)a,rti( ' le, th(; f()rc(; Fd t ~( teten ll i l f i~lg th e ra te of va,riat i( ) l l i1~ t i ~ e ( )f th e t )a .rt iclc ll l( ) ll le , l l t l l l l l p. W it ht im e ret)la.(',e(t l)y t,l~e ar(: ltu~gtl~ s a.~(t t h e ~tm(:lm,ifi(:al t)()to, ntia,1 })y tlw, ro, fra (: tiv ein de x '~,, t l~e gra (l i e~ t V 'n ~imy l)c i ld,( ;r t) r( ; t,e(1 a s t,l~( ; "f( )r( :e", a ,( ' t i~g ah m g th edrgra.(ti(nd, ()f th e t)(~t(ud, ial 'n( ir , y, z), a,~(t ' n ~ a,s th e l~(n~O ,l~t~ll~ (ff ti m tm,I'ti(:h'Y,w ho se ra te ( )f va,ria ,t i (~l 1)ci~lg ( teter~lfi~m(t t )y t l ie fi~r( :c V ~, [7].

Tl ie t, ral isi ti( ni if(nlx tl ie Im,gra,~lgia,ii t() th e IIa,xiiilt(niia,li l)i( :tt~re is th e nt )urmm(t ~n( t ( ; r Legend(Ire t ra ,~sf ( ) r~m,t i ( )n , whM ~ tak es . r ' a,~l( t y ' t( ) the r a y'm , m r p:~: m ~( l p:~ t, i r ()~ gt i

O L O LP:" 0 . , " ' P : ' / O y " ( 1 . 2 . 9 )

an(1 a,( '( :o r(lingly l,tl( '~ Im ,gr ml gim l fllll( 't i()xl t() t h e (q)ti(:a,1 tIa,lllill,()Iliml a,s~ 9 ( 1 .2 .10 )

By (1 .2 .5 ) t he ge l l e ra l i z ( ; ( l n lOI l l cn ta p : ,. a n d p , ext ) l i t : i t ly wri te a .s. r / .PX --II(;U, i l l , Z ) x/l_F:~2+y/2y'

p :, j - ' . ( : r , y , z ) v h + x ' ~ + . r ~

- - I t - -

- - - l t , ~

( t i l ;i t sd yd s '

( 1 .2 .11 )

in acc ord an ce w i th t i l e n l ( ;( : han i ( :a l ana lo gy sugg es t e ( t above ,. As we see, p xa n d p y e q u a l t h e l o c a l v a l u e ( )f t h e r e f r a c t i v e i n d e x t i m e s t h e d i r e c t i o n c o s i n e s( d.~ ~ ) o f t h e l i g h t r a y r e l a t i v e t o x a n d y a x i s, r e sp e c t iv e l y . He n c e , t h e y a r e--d;s , d sa l s o t e r m e d o p t i c a l d i r e c t i o n c o s i n e s of the ray .

W e e x p r e s s t h e f u n c t i o n ( 1 .2 . 10 ) i n t e r m s o f i ts p r o p e r v a r i a b l e s t o o b t a i nt h e H a m i l t o n i a n o f t h e r a y in t h e f o r m

y , - - y , - - ( 1 . 2 . 1 2 )

w h i c h i s j u s t t h e n e g a t i v e o f t h e o p t i c a l d i r e c t i o n c o s i n e o f t h e r a y a l o n g t h ez d i r e c t i o n .


7/58

H a m i l t o n i a n P i c t u r e o f L i g h t O p t ic s . F i r s t - O r de r a y O p t i c s 7

F i n a l l y , i n a n a l o g y w i t h m e c h a n i c s w e m a y w r i t e H a m i l t o n ' s e q u a t i o n s f o rth e l i g h t r a y a s d q x O H d q v O H

_ ~ _ . _ _d z O p x ' d z O p ydpx OH dpy OH ( 1 . 2 . 1 3 )d z Oqx ' d z Oqy '

t h u s c o m p l e t i n g t il e H a m i l t o n i a n f o r m u l a t i o n o f g e o m e t r ic a l o p t ic s . W e c h a n g e dx --+ qx and y ~ qy f or p u r p o s e o f f u t u r e u s e.

E v i d e n t l y , w i t h i n t h e m e c h a n i c a l a n a l o g y t h e l i g h t p r o p a g a t i o n c a n b e a s -s i m i l a te d t o a d y n a m i c a l p r o b l e m w i t h t w o d e g r e e s o f f re e d o m , t h e l e n g t hc o o r d i n a t e z b e i n g g i v e n t h e d i s t i n g u i s h e d r o l e a s t h e i n d e p e n d e n t ( e v o l u t i o n )v a r i a b le . W e e m p h a s i z e t h a t t h e z d i r e c t i o n c a n b e a r b i t r a r i l y f ix e d , a n d h e n c ec o n v e n i e n t l y c h o s e n t o c o in c i d e w i t h s o m e p r iv i l e g e d d i r e c t i o n o f t h e s y s t e m .O r d i n a r y o p t i c a l s y s t e m s a r e c o m m o n l y d e s i g n e d t o h a ve a p l a n e o f s y m m e -t r y , w h e r e a n i dea l r a y c a n b e i d e n t i fi e d , w h i c h r e p r e s e n t s t i l e p a t h o f a r a yt h r o u g h t i l e s y s t e m c o r r e s p o n d i n g t o a d e q u a t e l y a s s i g n e d i n p u t c o n d i t i o n s .T h i s ideal ray i s u n d e r s t o o d a s t h e opt ical axiso W e r e s tr i c t o u r a t t e n t i o n t ocen tered o p t i c a l s y s t e m s , w h e r e a l l o f t h e o p t i c a l c o m p o n e n t s a r e a li g n e d w i t ht h e i r o p t i c a l a x e s l y i n g o n t h e s a m e s t r a , i g h t l i n e , w h i c h t h e n m a y b e d e s i g -n a , t e d a s t h e o p t i c a l a x i s f o r t h e o v e r a l l sy s t e m; t h e z a x i s ma y b e c h o se n a , st h i s c o m m o n a x i s . T h e o ptic a,1 a x i s o f a sy s t e m o f c o a x i a l l e n se s, f o r i n s t a n c e ,i s j u s t t i le C Ol lltilO i l a x i s o f t i le l e n se s a n d c a n c o n v e n ie n t ly b e t a k e n a s z a x i s .T h r o u g h o u t t h e b o o k w e r e f e r t o t h e o p t i c a l a x i s a s t h e r e f e r e n c e z a x i s .

E q u a t i o n s ( 1 . 2. 13 ) p r o v i d e t i le f o rm a l a n s w e r t o t h e b a s i c p r o b l e m o f g e -o l n e t r i c a l o p t i c s o f d e t e r m i n i n g t h e f i n al v a r i a b l e s o f t h e r a y a f t e r p a s s i n gt h r o u g h a n o p t i c a l s y s t e m , o n c e a s s i g n e d t h e r a y v a r i a b l e s b e f o r e p r o p a g a t i o na n d sp e c i fi e d t h e o p t i c a l p r o p e r t i e s o f t h e s y s t e m . I n f a c t , g iv e n t h e v a, u e s o ft h e p o s i t i o n a n d d i r e c t i o n c o o r d i n a t e s o f t h e r a y a t s o m e z i, i. e. , t h e i n i ti a lv a lu e s f o r t h e e q u a t io n s o f m o t io n (1 . 2 . 1 3 ) , i t i s p o s s ib l e i n p r in c ip l e t o so lv e( 1 .2 .1 3 ) f o r t h e r a y p o s i t i o n a n d d i r e c t i o n c o o r d i n a t e s a t a n y o t h e r z .

F o l lo w i n g a g a i n t h e s u g g e s t io n o f H a m i l t o n i a n m e c h a n i c s , w e c a n c o n s t r u c tt h e g eo m e t r i c a l -o p t i c a l p h a s e s p a c e a s t h e C a r t e s i a n s p a c e o f t h e f o ur r a y -v a r i a b l e s ( q x , q y , p x , p y ) . N o t a b l y , a c c o r d i n g t o ( 1 . 2 . 1 1 ) , t h e r e f r a c t i v e i n d e x nr e p r e s e n t s t h e n o r m a l i z a t i o n f a c t o r t o t u r n t h e g e o m e t r i c a l v a r i a b l e s d q x / d sa n d d q v / d s i n t o t h e p h a s e - s p a c e v a r i a b l e s p x a n d p v . T h e c o r r e s p o n d i n g s t e pi n m e c h a n i c s i s t o re p l a c e ve l o c it y w i t h m o m e n t u m , t h e m a s s b e i n g t h e p r o p e rn o r m a l i z a t i o n f a c t o r .

I n t h e g e o m e t r i c a l o p t i c a l p h a s e s p a c e t h e l i g h t r a y i s i n d i v i d u a l i z e d b yt h e t r a j e c t o r y o f t h e p o i n t w h o s e z - e v o l v i ng c o o r d i n a t e s ( q ~ , q y , P z , P v ) f r o mt h e s p e c if ie d i n i t ia l v a l u e s a re d e t e r m i n e d b y t h e e q u a t i o n s o f m o t i o n . A c ol -l e c ti o n o f r a y s d i s t r i b u t e d o v e r a r a n g e o f p o s s i b l e p o s i t i o n s a n d d i r e c t i o n s


8/58

8 L i n e a r R a y a n d W a v e O p t ic s n P h a s e S p a c e

f il ls a, c e r t a i n r e g i o n i n p h a s e s p a c e , w i t h w h i c h w e c a n a s s o c i a t e a d e n s i t yd i s t r i b u t i o n f u n c t i o n p ( q . ~ , q y , p x , p y , z ) . T h e c h a n g e i n t h e r a y b u n d l e a s i tp r o p a g a t e s t h r o u g h o p t i c a l m e d i a r ef le ct s i n to t h e m o t i o n o f t h e r e p r e se n t a -t i v e p o i n t s a s t h e y m o v e t h r o u g h p h a s e s p a c e i n a ,c c o r d w i t h t h e e q u a t i o n so f m o t i o n ( 1 . 2. 1 3 ). W h i l e t h e e x a c t p h a s e - s p a c e m o t i o n o f e a c h r e p r e s e n t a t i v et )o in t ( i . e . , ca ,oh ray in the t ) lm( th~) i s un i ( tuc ly dc tc rmin ( ' , t by the in i t i a ,1 con-(tit, i()llS, it, is r a t h e r illlt)i'a,(:t,i(:a,|)le t o (:a,l(:lfla,t,e a,n exa ,(:t s()lllt, i()~l f()r t, tle w h o le1 )un ( l lc o f rays . I t i s thc re f ( ) rc co nv en ien t to 1 ) rov ide a, s ta ,t is ti ( :a ,1 des ( ' , r i t) t ionof t h e bc ha vi ol n" of t l lc a.sscllfl) ly of ra,ys, rt 'ga,I 'dC d a,s a, l l (uls(Ulfl) le ()f i ( lentica,1syst, cn ls ( t iffcI' i~lg ()ver a, rm lg e ()f in it i al ( : ()II( ti t, ( )IlS, an ti a( : ( : ( )r( t ingly t() follo wthe cvo l ld , ion ( ) f t,h (~ ( t e ns i t y ( l i s t r i l ) l l t i ( m f lm( : t i ( )n p .

(~ c( m lctr i c ( ) I ) t ics i s t ) r i~ lmri ly ( 'o~lcer~l (; (l wi th th e t )e ha vi ( nl r ( ) f a, s i l lg lc raywhen i t , t )a ,sscs t t l r ( ) l lg t l o t ) t i ( :a , l nmdia , . We wil l f i rs t ly ta , t 'k le t im g(ulera , l t ) ro l ) lcm( ) f the cvo l ld , i( )ll l ln ( t c r the I t a l l f i l ton im l (1 .2 .12 ) , in t ro ( t lu : i l lg t l l e 1 )a si ( : too l s( ) f t im Ha ,m il to l l -L ie f ( )n lm. l is ll l . T lm ll we wil l f ( ) t: l~s r a t t (ud, i ( )~ ( )~ l i~ma,rray -o t ) t i c s a .nd hen ( : e on t he re la , t cd ray-~na , t r ix fo rma ,l i sn~ . W e wi l l i lh~ s t ra tcth e a, | ) i l i ty ()f th e ~ m ,trix n~ct, ~()( t a ,s a, trm: ii~ g t() ( ) l ft )r | ) ( ) t l~ a, s i~ gl( ' I)a,raxia.1ra y a,n(t a,n (u~stu~fl) le ()f t )a ,raxia,1 ra ys , t ln~s tm .ral lelin g t lm sil~glt~ Im,rti t ' le an( tl)art i( : l t~-( ;nstu~flfle l ) i ( : t~res ( ) f ( :la.ssit :a .1 s ta , t i s t i t : a l met : tm,ni t: s . I ~ la te r (~ lmt) ters ,we wil l i l lus t ra t (~ t , l ,e s~f l )s ta ,** t ia] ly ( l i f feI ' (u , t t ( ) ( ) l s of l inear wavc-( ) t ) t i ( : s , ( lca l i l ,gw it h l ig ht ( t is tri | )~, t i ( )**s i**st('a(1 ()f | )~m(ll(~s ()f ra ys . Th(~ rc we wi ll i**tr()(l~,( :(~ t im' w a v e - o p t i c a l p h a . s ' ( ', , s 'p a ( ' ( : , wl~()s( ',


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Ham iltonian P icture o f Light Optics. First-Orde r Ray Optics 9o ' !

~ - !9 " " g l i is 0 ~ !. , . " ' 9 I n pu t p l an e i

. - -"" Ty ; !. s !s . S "

; ii z . , . ' "i i o,~-i s ~

9 s " li I ~ "

, ,. i . . , , ' ' ~ s "' . " ' ~ q x , q )i . . . " ' Y t

~ . ~ o ~*~. ~ " [ I o i .. s " I~ . . . - .~ O u t p u t p l a n e i-. Yo i

,'"'" ..1o ii ( q x ~ 1 7 6 i

ii! ~ . S ~

F I G U R E 1 .3 . A n o p t i c a l s y s t e m a n d t h e r e l e v a n t i n p u t a n d o u t p u t p l a n e s , to w h i c h th ec o o r d i n a t e s o f t h e i n c o m i n g a n d o u t g o i n g r a y s a re r e fe r r e d .

1.3 Ham iltonian picture of light-ray p ropagation" formalset t ingsW e c o n s i d e r a l ig h t r a y p r o p a g a t i n g a l o n g a n o ptic a.1 s y s t e m , w h i c h w e su p p o s et o b e c o m p l e t e l y c h a r a c t e r i z e d b y t h e r e f r a c t i v e i n d e x f u n c t i o n , g i v e n a t e a c hp o i n t b e f o r e a n d a f t e r t h e s y s t e m a n d i n t h e s y s t e m i t s e l f . T h e o p t i c a l z - a x i sis c o m m o n l y o r i e n t e d a s t h e d i r e c t i o n o f l ig h t p r o p a g a , t io n , a s s u m e d t o o c c u rf r o m l e f t t o r i g h t . T h e l i g h t r a y s t h e r e f o r e e n t e r t h e s y s t e m a t t h e l e f t a n de m e r g e f r o m i t a t t h e r i g h t . I t is n a t u r a l t o d e s c r i b e t h e e v o l u t io n o f t h e r a ya l ong t he ax i s by de t e rm i n i n g a,t any a,xia,1 pos i t i on t he i n t e r s ec t i o n po i n t o ft h e r a y w i t h a p l a n e o r t h o g o n a l t o t h e a x i s , a ,n d t h e o p t i c a l d i r e c t i o n c o s i n e so f t he r ay a t t ha t po i n t . Th us , we fi x t wo p l anes , I I i and I Io , ac ro s s t he o p t i ca lax i s , l oca t ed a t pos i t i ons z i a n d Zo, r e s p e c t i v e l y a t t h e l e f t a n d t h e r i g h t o ft he sy s t em ( F i g . 1 .3 ) . We wi l l r e fe r t o H i a s t he i n p u t p l ane and t o I Io a st h e o u t p u t p l a n e . A l s o , w e w i l l e q u i p e a c h p l a n e w i t h a c o o r d i n a t e s y s t e m . I nt h e f o l l o w i n g , u n l e s s o t h e r w i s e s p e c i f i e d , r e c t a n g u l a r c o o r d i n a t e s y s t e m s w i l lb e t a k e n i n b o t h t h e r e f e re n c e p la n e s . T h e o r i e n t a t i o n a n d t h e p o s i t i o n o f t h ea x e s c a n b e c h o s e n a r b i t r a r i l y ; i t i s a b a r e m a t t e r o f c o n v e n i e n c e t o c h o o s et i le o r ig i n s ly i ng on t il e op t i ca l z -ax i s and t he co o rd i na t e axes pa ra l l e l, a nde q u a l l y s c a l e d , f r o m o n e t o t h e o t h e r p l a n e .

T h e i n c o m i n g r a y i s u n a m b i g u o u s l y i d e n t i f i e d b y t h e c o o r d i n a t e s ( q ,~ , q w )o f t h e i n t e r s e c t i o n p o i n t w i t h t h e i n p u t p l a n e I I i , a n d b y t h e o p t i c a l d i r e c t i o ncos i nes (Px~ , Py~) a t t h a t p o i n t . E x p l i c it ly , w e c a n w r i t e

Px i - - n i sin ~x~ , py~ - - n i s in ~y~, (1 .3 .1)


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1 0 L i n ea r R ay a n d W a v e O p t i c s n P h a s e 5 p a c e

q x q " lr a y s e c t i o n " ~ . ." ...

i ~ . . , o , . . " . . . .

" ' " ' " " . s e c t i o n " ' " ' "(a) (b)

F I G U R E 1 .4 . T h e aI l gl c s (a) r~. a l~( l (b) rh, , c l l t c . r il l g t i l e ( l c f i ll i ti o x l ( )f t i l e ( ) p t i c a l l n o m e n t ap , . a n ( l p .~ ,, a r c t h e ( ' ox * * pl c* * m ,d ,s t o 2 o f t i m m , g l c s ( ) f t i m r a y t o t i m ; r a , , ( l ! / ; t x i s , r e s p e c t i v e l y .

w hc r( , 'n,~ ~_ 'n,(q.,:~, q:,~,, z i) is tl~( : l()(:al va,l~t(; ()f tl~(; rcfl'a (:tiv (; i~r a t th e in tc r-sc( : t ion I)oin t m~ (I ( (Lr,, (~ 'U,) axe l ,l~( ; a .~gh;s, t l~( ; i~ ( : ( )~ i~ g ray ( lcfi l~( ;s w ith th eassign(;(l t)la,n(;s y i - z a l~ (l . r i - z (F ig. 1.4) . Sin fi larly , th( ; ( )~tl ,g() i~g ray is st )c( : i f icd|)y the coor(li~m,tt;s (q.r,,, q.q,,) of l,|~(; i~d,crsccl, i{)~ w it h I,l~(; t)~l,l)~t, t)la~(; IIo an til) y I ,l ~ ( ; ( ) I ) l , i ( : a l ( l i r t ; ( ' l , i { ) ~ ( ' . { ) s i ~ c s ( p . , . , p ~ / , , ) a,t l,l~is I)lm m:

P . r , , - - 1 } o S i l l ~ k r , , , p y , , - - l t o S i l l ( ~ y , , (1 .3 .2)whc l'C '~.o ~ 'n(q , : , , , qv , , , z , , ) a.~r (,~:,,,,, ~:,/ ,) ar c tl~(; a.I~gl(;s ()f tll( : ra,y I 'cla tivc tot h e t ) l a n c s yo- z a ll< t . r , , - z .

E v id en tl y , tile t)osil, i()ll ( :o( ~r( lillatcs q:,. an(1 qv llavc th e (ti~ll(;llSi()llS ()f le ng th ,w hi ls t th e (tirc (:t i(m (:~) ()r(lilmtcs p.,. a n(t Pu a.r( '~ (lilliensi()~fl(;ss.

A s a, r('~a.(ly ~()t,('~(1, t,l~c t)r( )l) h;l ~ (~f g('~()~('4,ri(:a.1 ()t)I,i('s is l,(~ (I( ;t(; n~ finc t h eint )u t -o ~ t t ) ~ t r ( ; la t i (m s ( ) f th e oI )t i( :a,1 syst( ;n~,

q:ro = q.~-, q,r,, qv;, P.,:,, P.,j, ; z ,, ),qyo = qyo ( q:,:, , q:q;, P.r, , Pv, ; z o ) ,

p , , o = p . ,: ~ , ( q , , , ~ , q y , , p , : , , p : , j , ; z o ) ,P u o = p , . , / . ( q . ,. ~ , q v ~ , P : ,: .~ P v . ~ ; z , ) ,

which re l a t e in a ( t c s i ra , | ) l c cx t ) l i ( : i t fo rm t i l e t ) os i t i on and d i re ( : t i (m ( : ( ) o rd ina te so f t h e i n c o m i n g r a y a t z i t o t h o se o f t il e c o r r e s t ) o n d i n g o u t g o i n g r a y a,t Zo.

T i l e i n p u t - o u t p u t r e l a t i o n s ( 1 . 3. 3 ) d e s c r ib e a t r a n s f o r m a t i o n i n t h e o p t i-(:a,1 pha se-s pa ,c e of t i le rep res en ta , t i ve po int Pi --= (q :r , , qy , , Pz~, Py,~) of th e in pu tray to t l l e r epresen ta t ive po in t Po - - - - ( q zo , qyo , PZo , Pyo) o f t h e o u t p u t r ay . E v i-d e n tl y , th e t r a n s f o r m a t i o n o c c u rs t h r o u g h a t r a j e c t o r y f r o m P i to P o, r u l e d b yt h e e q u a t i o n s o f m o t i o n ( 1 .2 .1 3 ) w i t h t h e i n h e r e n t H a m i l t o n i a n ( 1 .2 .1 2 ) . T h et r a n s f o r m a t i o n ~ > Po is s y m p l e c t i c .


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Ham iltonian P icture of L ight Optics. First-Order Ray Optics 11

1 . 3 . 1 R a y p r o p a g a t i o n a s a s y m p l e c t i c t r a n s f o r m a t i o nW e c la r i f y t h e f o r m a l a ,n d p h y s i c a l s i g n i f i c an c e o f t h e r a y p r o p a g a t i o n a s as y m p l e c t i c t r a n s f o r m a t i o n [3, 9].

T o t h i s e n d w e f i r s tl y n o t e t h a t , l ik e t h e m e c h a n i c a l p h a s e s p a c e , t h e o p t i c a lp h a s e s p a c e n a t u r a l l y c o n f ig u r e s a s a 4 D l in e a r v e c t o r s pa c e, t h e 4 - v e c t o r s ub e i n g f o rm e d b y t h e z - d e p e n d i n g r a y - c o o r d i n a t e s ( q x , q y , P x , P y ) . S i m i l a r l y i tc a n c o n v e n i e n t l y b e e q u i p p e d w i t h t h e e u c l i d e a n m e t r i c d e f in e d a s u s u a l b yt h e s c a l a r p r o d u c t o f v e c to r s "

4( u , v ) - ~ u j v j - u T - v , ( 1 . 3 . 4 )j = l

w h e r e u n- is t h e t r a n s p o s e v e c t o r a n d , f o r n o t a t i o n a l c o n v e n i e n c e , t h e v e c t o rc o m p o n e n t s a re i n d e x e d b y t h e i n te g e rs f r o m 1 t o 4 . C o n f o r m i n g w i t h t h ec o m m o n p r a c t ic e , b y " v e c t o r s " w e m e a n c o l u m n v e c t o r s ; t h e n , ro w v e c t o r sa r e id e n t i f ie d b y t r a n s p o s i t i o n , i .e ., u - ( q x, % , p ~ , p y ) - 7 _ (Ul , u2 , u3 , u4 ) r .

E x p l o i t i n g t h e v e c to r n o t a t i o n , w e c a n w r i t e H a m i l t o n ' s e q u a t i o n s ( 1 .2 . 13 )i n o n e f o r m d u j 4 O Hd z = ~ , lj l Ou l ' j - 1, . . , 4, (1 .3. 5)

l = 1w i t h t h e a i( t a s we l l o f t i l e su i t a b l y ( t e fi n e d c o e f f i c ie n t s J j l , wh i c h a, e a l l o w e dt o b e o n l y 1 , - 1 o r 0 a c c o r d i n g t o t h e i r o w n i n d i c e s "

, ] j l - ~ i l , j + 2 , j - 1 , 2 ; , ] j l - - ~ l , j - 2 , j - 3 , 4 . ( 1 . 3 . 6 )E v i d e n t l y , J j l ' s c a ,n c o n v e n i e n t l y b e a , r r a ,n g e d i n t o t h e 4 x 4 a ,n t i s y m m e t r i cm a t r i x J as (0I)J - - I 0 ' ( 1 . 3 .7 )wh e r e I d e n o t e s t h e 2 x 2 u n i t m a t r i x a s 0 t i l e 2 x 2 n u l l m a t r i x (b) . T h em a t r i x J is k n o w n a,s t h e s y m p l c c t i c u n i t m a t r i x . I t h a s t h e n i c e p r o p e r t i e s

j - 1 _ j r _ _ j , j 2 _ - I , d e t J - 1 , ( 1. 3.8 )j T d e n o t i n g t h e t r a n s p o s e m a t r ix .

W e s u p p o s e n o w t o t a k e t h e v e c t o r u t o t h e v e c t o r v - ( V l, v 2, V a, v 4 ) u n d e rt h e t r a n s f o r m a t i o n

V j - - V j ( U l , . . . , U 4 ) , j - 1, . . , 4 , ( 1 .3 .9 )b I t i s a c o m m o n c o n v e n t i o n t o r e p r e s e n t 2 m 2 m m a t r i c e s a s p a r t i t i o n e d i n t o f o u r m m

s u b m a t r i c e s [ 8 ] .


12/58

1 2 L i n e ar R a y an d W a v e O p t i c s n P h a s e S p a c e

i d e nt i fi e d b y t h e a ss o c i a t e d J a c o b i a n m a t r i x S , w h o s e e n tr i e s S i i = O v i / O u j( t e t ) en ( t in gene ra l on u j , j = 1, .., 4.

I t is c a. sy t o p r o v e t h a t v o b e y s e q u a . t i o n sO Hd v j = S : ) ~ , , I ~ , ~ ; S h a , j - 1, . . , 4, (1 .3. 10 )d z O v h

t l l ( ~, s ~ n ~ m t i ( ) ~ s i g ~ l s ( ) v ( w l " (: I) (; a,l ,( ;( 1 i ~ l( t i( : (: s t ) ( : i ~ g (mf i t t c ( t .W(; r( ; (p~ir (; th a t th( : t ra~ls f ( ) rl lmt i( )~l (1 .3 .9 ) h ;av( ;s t ll ( : f( )n~l ( )f Hanfi l l , ( )n 's

( ; ( t~ati()ns inva,rimd,, i . ( : . , t lmt v () t ) ( :ys th( ; san~(: ( : ( t~m,ti()IlS (1.3.5) a,s u, na,m(;lyd~ , j _ . l j OHd z - i)~ , t ' j - 1 , . . , 4 . (1 .3 .11)

W it ll t lfis r( ; ( t~u:st al) t )l i( ; (1 t( ) ( 1. 3. 1( )) , w(: ( ;~(1 ~1t) wi tl l t l l( : ~ m tr ix r( : la, t i(nl :S J S m - - J . ( 1 . 3 . 1 2 )

It r(~t)r(~s(~llts tll( ' ( '()ll( liti(nl f()r S t.() 1)(~ a syllll) l( '( ' ti(: ll m tr ix a.t rel y l)()illt inI)llas( : sI)a(:(; [9]. A s ll()t(:(l, ill fa (: t, S lll ay (l(:t)(;ll(l ill g(',21(;ral ()zi 1,11(; v(:(:t() r u .Ill ( '()ld,ra,s | ,, tll(: I)l '()(lll( 't S J S T r(:slll ts lil t( )t ll( ; lllatI'iX J , ill(l(:l)(:ll(l(:~d, ()f U.| ' ;vi( le .~d,ly tl l ( : l l l l i t l lm tr ix I m~ (l i l l( : l lm tr ix J a r( : syll lt ) l ( : ( : t i ( ' a s w (:l l.

W(: r(:(:a,ll tlla.t ill g(;l~(:ral ll(nlsi~ lgllla.r s(t~m,I'(: ~m .tri(:(:s fl ) r~ a. gr() ~ 1) ~nl(l(;rt | l(: r()w-(:()l~ lnlll ~ m tr i x ~11~ltit)li(:a, ti()~l, tl~(: (:xist(:~(:(: ()f (,11(: i(l(:~ til,y a n(l th e~nli(t~( : i~v(;rs( : as w(:l l as t l~( : ass () ( ' iat ivi ty 1)( :i~g a, I) lah~ ( : (n~s(:( l~uul( ' ( : ( )f th( ;~ m, trix ~n~ lti t) l i ( :al ,i ( )~ l)r() t ) ( :rti ( :s . I~ t)a ,rti ( :~la.r, l ,l~(: ~la. tri( : ( :s ( ) l ) ( ;yi~ g (1 .3.1 2)(1() f() rnl a. gr()~ q)(1~ (; t() tl~(: (:l()m~r('. I)r () t)c rty ()f tl~( ', sy~q )le,(:ti(: (:(n~(li| ,i()ll, wh i(: hix I)rcs( ;rv( :( l | ) y ~m l,rix ~n~ ltiI) l i ( 'a l ,i ( )~; i~ flu:t , if S l a~ (t S~ () | ) ( ;y (1 .3.1 2) , s ()wi l l th( ; I ) r( ) ( t~mt S~ S~. W(: i~vi t ( : t ll ( : rca , ( lcr t ( ) v( : r i fy t ha t , i f S is sy~ q)h ; ( ' t ic ,th e i i~v(:rsc S - 1 ex is ts a,n(t is syn~t) h;( : t ic a,s well a ,s th e tra,nst)()s( : S T. Ttl( : gro upof t h e 4 x 4 sy~ I)l (: (: ti(: r( ' ,al ~mt, i 'i( '( ;s ix (t(;~()t(;(t a.s S p ( 4 , R ) .

Sin ce ( le t J = 1 m~ ( t ( to t S = ( t (' .t S T , t )y ( 1 .3 .1 2) i t fo l low s th a t( l e t S = + 1 . (1 .3 .13 )

I t c a n b e s h o w n t h a t a c t u a l l y foI" a, s y m p l e c t i c m a t r i x i t is a l w a y s ( l e t S = + 1[.().2]. R em ar ka b l y , in ti l e 2 x 2 case t i l e nec es s a ry an d su f f i c ien t co nd i t io n fo ra m a t r i x t o b e s y m p l c c t i c is t h a t i t h a s d e t e r m i n a n t + 1 . C o n s e q u e n t t o t h es y m p l e c t i c c o n d i t i o n ( 1 . 3 . 1 2 ) , w e al s o fi n d t h a t t h e 1 6 e n t r i e s o f t h e m a t r i xS a r e n o t al l i n d e p e n d e n t ; o n l y 1 0 c a n b e a r b i t r a r i l y c h o s e n , t h e o t h e r s b e i n gf i x ed b y ( 1 . 3 .1 2 ) . T h i s r e f l ec t s t h e g e n e r a l p r o p e r t y o f t h e 2 m x 2 m s y m p l e c t i cm a t r i c e s , f o r w h i c h o n l y m ( 2 m + 1 ) o f t h e 4 m 2 e l e m e n t s c a n a r b i t r a r i l y b ea s s i g n e d ( P r o b l e m 1 ) .


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Ham iltonian Picture of Light Optics. First-Order Ray Optics 13

I n t e r e s t i n g l y t h e J a c o b i a n m a t r i x S r e l a t e s t h e i n f i n i t e s i m a l v e c t o r s d v a n dd u ( s e e a l s o P r o b l e m 2 ) . I n d e e d , b e i n g

4 O V i 4dv i - ~ ~ du j - E S i j du j , ( 1 . 3 . 1 4 )j = l j = lw e c a n c o n c i s e l y w r i t e

d v - S d u . ( 1 . 3 . 1 5 )I t is t h e r e f o r e a d i re c t c o n s e q u e n c e o f t h e s y m p l e c t i c c o n d i t i o n ( 1 .3 . 12 ) t h a t t h eq u a n t i t y ( d U l , J d u 2 ), fo r a n y c h o ic e o f d u 1 a n d d u 2 , r e m a i n s u n c h a n g e d u n d e rt h e s y m p l e c t i c t r a n s f o r m a t i o n ( 1 .3 .1 5 ) , w h i c h t a k e s u to v a n d a c c o r d i n g l y d u lto d v 1 a n d d u 2 t o d v 2. E x p l i c i t l y , w e c a n e a s i l y p r o v e t h a t

( d v 1 , J d v 2 ) = ( d U l , J d u 2 ) . (1.3.16)I n p a r t i c u l a r , i f t h e n e w a n d o l d v e c t o rs , v a n d u , a r c l i n k e d b y a l i n e a rt r a n s f o r m a t i o n a s vi - ~ = 1 a i j u j , i - 1 , . ., 4 , the en t r ie s o f S a re jus t th e u -i n d e p e n d e n t c o e ff ic i e n ts o f t h e t r a n s f o r m a t i o n , i .e ., S / j = aij . H e n c e , r e l a t i o n s(1.3.15) a n d (1.3.16) h o ld t r u e f o r t h e f i n i t e v e c to r s a s w e l l . Ex p l i c i t l y , i f

v = S u , ( 1 . 3 . 1 7 )w i t h S s y m p l c c t i c a n d u - i n d e p e n d e n t , t h e n

( V 1 , J v 2 ) = ( U l , Ju2). (1.3.18)I t is e v i d e n t t h a t i f t h e t r a n s f o r m a t i o n w h i c h t a k e s t h e v e c t o r u t o v is

z - d e p e n d e n t :V j - - V j ( U l , . . . , U4; Z) , j = 1, . ., 4, (1 .3.1 9)

t h e s i m p l e d e r i v a t i o n w e h a v e s k e t c h e d f o r t h e s y m p l e c t i c c o n d i t i o n ( 1 . 3. 1 2 )n o l o n g e r h o l d s. B u t , i n d e e d , t h i s i s j u s t t h e c a s e o f o u r m a i n i n t e r e s t . I n f a c t ,a s w c s a i d e a r l ie r , t h e p r o p a g a t i o n o f l ig h t r a y s t h r o u g h o p t i c a l s y s t e m s i sf o r m a l l y d e s c ri b e d b y t h e s o l u t io n t o H a m i l t o n ' s e q u a t i o n s ( 1 .3 .5 ) , w h ic h , f ora s s i g n e d i n p u t c o n d i t i o n s , y i e l d t h e r a y v a r i a b l e s , a n d s o t h e r e l e v a n t p h a s es p a c e v e c t o r u ( z ) , a t e a c h z a l o n g t h e o p t i c a l a x i s t h r o u g h t h e a x i a l i n t e r v a lw e a r e i n t e r e s t e d i n , s a y f r o m z i t o Zo. T h e r e f o r e , w h a t w e m u s t v e r if y is t h a tt h e r a y p r o p a g a t i o n f r o m z i t o Zo, w e m a y p i c t u r e a s a t r a n s f o r m a t i o n o f t h ei n i t i a l r a y - v e c t o r u ( z i ) t o t h e f i n a l r a y - v e c t o r U ( Z o ) :

u ( z g ) , U ( Z o ) = u ( z ~ + A z ) , A z = Z o - z ~ , ( 1 . 3 . 2 0 )r u l e d b y H a m i l t o n ' s e q u a t i o n s ( 1 .3 .5 ) , b e s y m p l e c t i c .


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14 L i n e a r R a y a n d W a v e O p t i c s n P h a s e 5 p a c e

T o p r ov e t h is , i t is e n o u g h t o d e m o n s t r a t e t h a t a n y i n f i n it e s im a l t r a n s fo r -m a t i o n a h m g t h e p a t h ( 1 . 3 .2 0 ) i s s y m p l e c t i c , i .e ., t h e r e l e v a n t J a . c o b i a n m a t r i xo b e y s r e l a t i o n ( 1 .3 . 12 ) . T h e c o n t i n u o u s e v o l u t io n o f t h e r a y v e c to r u ( z ) a l o n gthe f in i t e space in te rva l A z c an be v iew ed as a success ion o f in fin it esima ,1 sym -t ) lec ti ( : t ra ns fo rm at i on s , wh ich , in v i r tue o f the p ro (h lc t t ) rese rva . tion t) r ( ) t ) c rtymen t i (n ic( t a , t)ovc, res ul t in to a . s ingle f in i te symt) lc( : ti ( : t ran sfo rm at io i l .

T he li, let ~lS (:o~si(l( 'x a,n iI~finit(;si~m,1 tra,nsf()r~ m,ti()~ a,long th e (;v()h~tiont)a,t l~ (1.3.20 ) l ,aki~g th(; i~fi tial ray v(;(: t( )r u(z,~) at zs t( ) t l~( ; r ay vc(: t ( )r u ( z i + d z )at th e i l lfinit( ;s i~ mlly ( : l ( )sc axia,1 t)()si t i (m z i + d z . F ( ) r (:()nv(;nien(:(;, w(: set

w - u ( i)du I d z- u ( z i + d z ) - w + ~ : , (1 .3 .21)

wh (;r c, ()f (:()In's(;, ()n ly tirst- () r( lcr tc rn ls in t he (;l( '~nlclltal (list)la.(:( ' .nl(;nt d z have})(;(;n r('.taill('~(l.

In a(:(:()r(l w itll H a,lllilt (m 's (;(tlmti()llS (1 .3.5 ), ()ll( '~ ll~LSd u jd z

0 t i- - , l j lZi

(1 .3 .22)

wlf icll, (hi(:( ; ills('rt,(',(t illt,() tll(' s(;(:()~l(l ()f (1.3 .2(} ), yi( 'l(lsO H d z . ( 1 . 3 . 2 3 )

~."7

Tll(,~ ,Ja,(:()])ia,]l ll m tr ix (:(ram s t() t)(;Ot,j Ou~j 0 2H

S j l ..... O'u '! ..... O 'u '! _of ,]jl,: Ou lOU l,: (1 .3 .24 )

wh ere the se ( :ond o r ( to t der i va t iv es o f t i l e H am i l t on ian f lmct ion , H[~ . - o ,, ,ouk z~w hich forn l t im s () -( :a ,lh ;(1 Hcss ia ,n ma tr ix H " - (H [ ~ )~ k , a,rc cva,lua,tcd a,t u(zi)a n d z i . D l u ; t o t h e a s s lH n c d i n t e r c h a n g e a b i l i t y o f t h e d e r i v a t i v e o rd e r w i t hre s p e c t t o t h e v e c t o r c o m p o n e n t s i n t h e H e s s i a i ~ e n t r i e s : H i '~ - H ~ , r e l a t io n( 1 . 3 . 2 4 ) ( : a n f i n a l l y b e c a s t i n t h e m a t r i x fo rm

S - I + J H " ( w , z i ) d z . (1 .3 .25)I n o r d e r t o a s se s s t h e s y m p l e c t i c i t y o f S w e m u s t t a k e t h e p r o d u c t S J S q-w h i c h , c o n s i s t e n t l y t o f ir s t o rd e r , i s e a s i l y p ro v e d t o b e :

S J S T - ( I + J H " ( w , z i ) d z ) J ( I - H " ( w , z i ) J d z )= J + J H " ( w , z i ) J d z - J H " ( w , z i ) J d z - J . (1 .3 .26 )


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Hamiltonian Picture of Ligh t Optics. First-Order Ray Optics 15T h e s y m p l e c t i c i t y of t h e i n f in i te s im a l t r a n s f o r m a t i o n m a p p i n g t h e i n i ti a l

r ay ve c t o r u ( z~ ) t o t h e i n f in i te s i m a , ll y d i sp l aced vec t o r u ( z~ + d z ) a l o n g t h ef in i te e v o l u t io n p a t h ( 1 .3 .2 0 ) g o v e r n e d b y H a m i l t o n ' s e q u a t i o n ( 1 .3 .5 ) i s t h e nd e m o n s t r a t e d . A c c o r di n g ly , w e m a y t h i n k o f t h e r a y p r o p a g a t i o n a s a c o n ti n -u o u s s e q u e n c e o f s y m p l e c t i c t r a n s f o r m a t i o n s , u n d e r w h i c h t h e r a y v e c t o r s inp h a s e s p a c e a r c t a k e n o n e t o t h e o t h e r , t h e a x i a l c o o r d i n a t e z b e i n g t h e s t e p -p i n g p a r a m e t e r . W e m a y s a y t h a t o p t ic a l s y s te m s o p e r a t e s y m p l e c t ic t r a n s -fo rm a t i ons on t he r ay va r i ab l es q and p [10 ] .

I t is w o r t h e m p h a s i z i n g t h a t t h e f o r e g o i n g d i sc u s s io n , w e h a v e d e v e l o p e dw i t h s p e c i f i c r e f e r e n c e t o t h e l i g h t r a y p r o p a g a t i o n , f r a m e s w i t h i n t h e g e n e r a li ss u e o f H a m i l t o n i a n m e c h a n i c s [ 3], s t a t i n g t h a t a n y canonical t r a n s f o r m a t i o n( i. e. , s u c h t o m a p p h a s e - s p a c e v e c t o r s o b e y i n g e q u a t i o n s o f m o t i o n o f H a m i l -t o n i a n f o r m i n t o p h a s e - s p a c e v e c t o r s o b e y i n g a s w el l e q u a t i o n s o f m o t i o n o fH a m i l t o n i a n f o r m ) i s s y m p l c c t i c , w h e t h e r i t d e p e n d s o r n o t o n t h e e v o l u t i o nv a r i a b l e i n h e r e n t i n t h e d y n a m i c a l p r o b l e m u n d e r e x a m i n a t i o n .

R e a d e r s i n t e r e s t e d i n a r i g o ro u s t r e a t m e n t o f s y m p l e c t i c g r o u p s a r e d i r e c t e dt o [ 9]. A d d i t i o n a l c o m m e n t s a r e g iv e n in C h a p t e r s 2 a n d 3 . T h e A p p e n d i x p r o -v i d es a n e l e m e n t a r y e x p o s i t io n o f t h e b a s i c n o t i o n s p e r t a i n i n g t o L i e a l g e b r a sand L i e g roups .1 . 3 . 2 P o i s s o n b r a c k e ts a n d L i e o p e r a to r sC e n t r a l t o o u r a p p r o a c h t o th e g e o m e t r i c a l o p t i c s p r o b l e m o f d e t e r i n i n i n g t h et r a n s f o r m a t i o n ( 1 .3 .3 ) o f t h e l i g h t - r a y c o o r d i n a t e s , g e n e r a t e d b y ti le H a m i l t o -n i an ( 1 .2 .10 ) t h rough Eqs . ( 1 .2 .13 ) , a r e t he s t r i c t l y r e l a t ed concep t s o f Pois-son bracke t a n d Lie operator. P o i s s o n b r a c k e t s a n d L i e o p e r a t o r s r e p r e s e n tt h e f o r m a l to o l s to i n t r o d u c e t h e H a m i l t o n i a n e q u a t i o n s o f m o t i o n i n t o th ea p p r o p r i a t e L i e a l g e b r a i c a m b i e n c e , w h e r e t h e i r i n t r i n s i c s y m m e t r y c a n m o r ec l ea r l y be ev i denced and e f f ec t i ve l y exp l o i t ed . The de f i n i t i ons and m a i n p rop -e r t ie s o f b o t h t h e P o i s s o n b r a c k e t s a n d L i e o p e r a t o r s a r e r e ca l l ed b e l ow . T h ed i s cus s i on i s deve l op ed a t a ge ne ra l l evel . Thus , ( q, p ) deno t es a s e t o f canon i -c a l l y c o n j u g a t e v a r i a b l e s f o r s o m e H a I n i l t o n i a n s y s t e m a n d z t h e i n d e p e i l d e n tv a r i a b le , i n d e p e n d e n t l y o f t h e i r s p e ci fi c m e a n i n g i n ra y o p t i c s.Poisson bracke tsL e t f a n d 9 b e a n y t w o c o n t i n u o u s l y d i f f e re n t i ab l e f u n c t i o n s o f t h e c a n o n i c a lva r i ab l es ( q , p ) . Th e P o i s son b rac ke t o f f and g , de no t ed by { f , g } , is s t il lf u n c t i o n o f ( q , p ) , fo rm ed ac co rd ing to [3, 10.2, 11]

Of Og{ f ' g} =~-~ Oq~ Op~OL O f Og "~ (1 .3 .27)Op,~ Oq~ J '


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1 6 L i n e a r R a y a n d W a v e O p t ic s n P h a s e 5 p a c e

w h e r e t i le s u m r u n s o v e r al l t il e p a i r s o f c o n j u g a t e v a r i a b l e s ( q ~ , p ~ ) fo r t hep r o b l e m a t h a n d . S i n c e b o t h t h e f u n c t io n s f a n d g m a y d e p e n d o n t h e i n de -p e n d e n t v a r i a b l e z , t h e P o i s s o n b r a c k e t { f , g } m a y d e p e n d o n z a s w ell.

C l e a r l y t h e P o i s s o n b r a ( ' k e t o t) e ra , ti o n c r e a t e s a b i n a r y r e l a t io n : ( f , 9 ){ f , g} o f e v e r y p a i r o f d y im ,m i ( :a l v a r ia .b le s t o o n e d y n a m i c a l v a r i a b l e , i d e n t i f i e dt h r o ~@ l t h e r u l e ( 1 .3 .2 7 ) . I n p a r t i ( : ~ f l a r , t h e r e l a t i o i ~ s

{ q , , , p / ~ } = ( 5~ ,~ ; ~ , ( 1 .3 .28 )( :hara( : terize the ( :m~()ni( :a .l ly ( : (m.j~ga.te wtriat) les ( q , ~ , p , , ) , as the f ( ) l l ( )wing

i ) f O . f{ q " ' " f } - i ~ ' p ,~ ' { P " ' " f } - O q , , ' ( 1 . 3 . 2 9 )esta .t)lis h tll( ; rlll(~s I,() tin'ill l ,lle P() iss( m t) ra( :k ets ()f aaly tm.ir ()f ( :(n ljllga.tc va ri-a.1)h's ( q , , , p , , ) witl~ a fi~t:l ,i(n~ .f ()f ( q , p ) . In t )a ,rt i t :~l lar, i f .f is ta .ke~ as thetIa,l l f i l to~ia,~ fl~xmti(m H, t l~c l ) (fisso~ t )ra,( :kets (1.3.29) rcI)r( ) ( l~( :e t t~e rightlm,~(t s i ( les of Ha,~i l t , ( )~ 's ( ' . ( l~m,t , i ( )~s , wl~i( ' t~ a , ( : ( : ( ) I ' ( l i~g ly w r i t e a l s o a.s

d q , , d p , ,d z = { q ' ' ' I I } , d z = { p ' ' ' H } ' ( 1.3 .3 0)f~)r ( 'v ery t )air ( )f ( : ( ) I l j l lgat( ' varial) l ( ;s ( q , , , p , , ) .

E vi (h ;n tly th e P() iss(m t)ra.(:k(; t () f .f and th( , v ( ,( : tor-va.l lw(l f ll~l( :t , t )n g( .q , ( q , p ) , . . , g . . . . ( q , p ) ) Y is t i le ve( : t ( )r f l l l l ( : ( , i ( ) I I { f , g } = ( { f , . q , } , .., {f,.q,,, } ) TT h e Po is s( m |)ra


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Ham iltonian Picture of Light Optics. First-Orde r Ray Optics 17

r e a d e r m a y e a s il y v er if y, t h e q u a d r a t i c h o m o g e n e o u s p o l y n o m i a l s o f t h e t y p ea q 2 + b p 2 + c q p a r e c l o s e d u n d e r P o i s s o n b r a c k e t s a n d s o t h e y s p a n a L i e a l g e b r a .T h e q u a d r a t i c h o m o g e n e o u s p o l y n o m i a l s i d e n t i f y a p a r t i c u l a r l y i n t e r e s t i n gs u b s e t w i t h i n t h e o v e r al l sp a c e o f d y n a m i c a l v a r i a b l e s { f ( q , p ) } , s i n c e t h e y a r eb a s i c t o t h e r e p r e s e n t a t i o n o f l in e a r o p t i c a l s y s t e m s i n b o t h g e o m e t r i c a l a n dw a v e o p t i c s . I n f a c t , a s w e w i l l s e e , i n t h e l i n e a r a p p r o x i m a t i o n t h e o p t i c a lH a m i l t o n i a n ( 1 . 2 . 1 2 ) t u r n s i n t o a q u a d r a t i c p o l y n o l n i a l . W e a l s o n o t e t h a t t h es u b s e t o f t h e i n h o m o g e n e o u s l i n ea r p o l y n o m i a l s o f t h e t y p e a q + b p + c d o e sf o r m a L i e a l g e b r a a s w e l l ; i n p a r t i c u l a r , t h e P o i s s o n b r a c k e t o p e r a t i o n i m a g e st h i s s u b s e t o n t o t h a t o f c o n s t a n t f u n c t io n s . I n C h a p t e r 4 w e w i ll se e t h a tb y th e " q u a n t i z a t i o n " p r o c e d u r e t h e L ie a l g e b r a o f t h e i n h o m o g e n e o u s l in e a rp o l y n o m i a l s m a p s t o t h e H e i s e n b e r g - W e y l a l g e b r a , w h i c h w i l l a p p e a r t o b e t h eb a s ic b u i l d i n g u n i t o f w a v e o p ti c s, i n a c c o r d a n c e w i t h t h e c l a i m e d a n a l o g y o fw a v e o p t i c s t o q u a n t u m m e c h a n i cs . A l so , t h e q u a d r a t i c H a m i l t o n i a n f u n c t i o n so f r a y o p ti c s w il l m a p t o t h e q u a d r a t i c H a m i l t o n i a n o p e r a t o r s o f w a v e o p t i c sa n d c o r r e s p o n d i n g l y t h e L i e g r o u p o f t h e s y m p l e c t i c m a t r i c e s o f r a y o p t i cs w i llm a p t o t h e m e t a p l e c t i c g r o u p o f t h e F r e s n e l - in t e g r a l o p e r a t o r s o f w a v e o pt ic s .

F i n a l l y , w e e s t a b l i s h a f u r t h e r c h a r a c t e r i z a ti o i ~ o f t h e s y t n p l e c t i c t r a n s f o r -m a t i o n s i n t e r m s o f t h e P o i s s o n b r a c k e t s . W e re f e r t o t h e 4 D t ) h a s e s t) a c e o fg e o m e t r i c a l o p t i c s a n d g a t h e r t h e r a y v a r i a b l e s i n t o t h e 4 - v e c t o r s u , a,s i n w1 . 3 . 1 . I t i s e a s i l y s e e n t h a t t h e s y m p l e c t i c u n i t m a t r i x J d i r e c t l y r e l a t e s t o t h ef u n d a l n e n t a l P o i s s o n b r a c k e t s ( 1 . 3 . 2 8 ) , w h i c h s y n t h e s i z e i n o n e f o r m

{ u i , u j } = J i j , (1 .3 .32 )a s w e l l a s t o t h e g e n e r a l P o i s s o n b r a c k e t ( 1 . 3 . 2 7 ) , w h i c h w r i t e s c o n c i s e l y a s

O f O g{. f , . q} - ~ J i j 9 (1 .3 .33 )' O U jC o m p o s i n g t h e d e r i v a t i v e s ( O f ) i = 1 ,4 a n d ( ~ ) i = 1 ,4 i n to t h e 4 - v e ct o rs O,~ f01Z i ,.. . ,'"a n d Ou9 , t h e r i g h t - h a n d s i d e o f ( 1 .3 .3 3 ) c a n b e i n t e r p r e t e d a s t h e s c a l a r p r o d u c to f t h e v e c t o r s O ~ f a n d J O ~ g . I n s y m b o l s w e w r i t e

{ f , g } = ( O ~ f , J O ~ g ) . (1 .3 .34 )L e t t h e p h a s e s p a c e v e c t o r u t r a n s f o r m t o v u n d e r a s y m p l e c t i c t r a n s f o r m a -

t i o n , w h o s e a s s o c i a t e d J a c o b i a n m a t r i x S o b e y s t h e r e b y t h e r e l a t i o n ( 1 . 3 . 1 2 ) .B y th e c h a i n ru l e f or d e r i v at i v e s, o n e h a s t h a t 0~ = ( S - r ) - 1 0 , . T h e n , r e g a r d i n gf a n d g a s f u n c t i o n s o f t h e n e w v a r i a b l e s v , w e f i nd

( O , f , J O v 9 ) - ( O u f , s - l j ( s T ) - l O u g ) - ( O u f , J O u g ) , (1 .3 .35 )w h i c h s h ow s t h a t t h e P o i s s o n b r a c k e t { f , g } h a s n o t c h a n g e d u n d e r t h e s y m -p l e c ti c t r a n s f o r m a t i o n u - -- , v ( u ) . C o n v e rs e ly , i f t h e i d e n t i t y ( 1 .3 .3 5 ) h o l d s,t h e n t h e m a t r i x S o f t h e t r a n s f o r m a t i o n u ~ v ( u ) is s y m p l c c t i c .


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1 8 L i n e ar R a y a n d W a v e O p t i c s n P h a s e S p a c e

C o n s e q u e n t l y t h e n e c e s s a r y a n d s u ff ic i en t c o n d i t i o n f o r a t r a n s f o r m a t i o n t ob e s y m p l e c t i c is t h a t i t p r e s e r v e s t h e P o i s s o n b r a c k e t s ( 1 . 3 .2 7 ) o f d y n a m i c a lv a r i a b le s a n d i n p a r t i c u l a r t h e f l m ( l a m e n t a l P o i s s o n b r a c k e t s ( 1 .3 .2 8) . T h ei nv a ,r ia n ce o f t h e P o i s s o n b r a c k e t s u n d e r s y m p l e c t i c t r a n s f o r m a t i o n s c o n f o rm sw i t h t h e i n v a . ri a n c e o f t h e s k e w p r o d u c t a.s s t a . te d i n ( 1 . 3 . 1 6 ) .L i e o p er a t o r sL r f ( q , p , z ) 1 )e s ( ) l i i e ( : () li t, i l u l ~ ) l l s l y ( l i f f e r r f l u l ( : t i ~ ) l i ~ ) f t , l ~ ; ( :~) l l . i~lgatev ar ia l ) l e s q , p . W e a~s () (: ia, l, r w i t i i f l , h r Li< ' O p f ' , ' F f L ~ , 0 7 " L / t ) y t h e l u l l e [ 1 1 , 4 ]

Lf ( q , p , z ) - { . , f } =~-~ , i~q


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Ham iltonian Picture of Ligh t O ptics. First-Order Ray Optics 1 9

L i k e t h e P o i s s o n b r a c k e t , t h e L i e o p e r a t o r a c t s o n t h e o r d i n a r y a n d P o i s s o np r o d u c t o f f u n c t i o n s in t h e s i m i la r m a n n e r a s th e d e r i v a t i v e o p e r a t o r , a n dh e n c e t h r o u g h a L e i b n i t z - l i k e r u l e .

W e c o n s i d e r n o w t h e s e t o f d y n a m i c a l v a r i a b l e s { f ( q , p ) } , wh i ch , a s p rev i -o u s l y r e m a r k e d , c o n f i g u r e s a s a L i e a l g e b r a u n d e r t h e P o i s s o n b r a c k e t o p e r a -t io n . E v i d e n t l y , b y ( 1 .3 .3 6 ) w e c a n i n d i v i d u a l i z e a c o r r e s p o n d e n c e o f { f ( q , p ) }

At o t h e s e t o f t h e a s s o c i a t e d L i e o p e r a t o r s { k i } "A{ f ( q , p ) } ~ { k f } . ( 1 . 3 . 4 0 )

T h i s s e t c a n b e s t r u c t u r e d a s a l i n e a r s p a c e ; i n f a c t , e x p l o i t i n g t h e l i n e a r i t y o ft h e P o i s so n b r a c k e t o p e r a t i o n , w e d e f in e l in e a r c o m b i n a t i o n s o f L i e o p e r a t o r sb y t h e r u l e

A A A

aL l + bkg - kaf+ bg - - { ' , a f + b g } , (1 .3 .41)t o ob t a i n ye t a L i e ope ra t o r .AN o t a b l y , t h e o p e r a t o r s { k l } s p a n a L ic a l g e b r a w i t h r e s p e c t t o t h e c o m -

m u t a t i o n b e t w e e n o p e r a t o rs . W e r ec a ll t h a t t h e c o m m u t a t o r o f t w o o p e r a t o r sA AF a n d G is d e f in e d b y t h e a n t i s y i n m e t r i c p r o d u c t

A A A A A A

I F , G ] - F G - G F . ( 1 . 3 . 4 2 )AT h e d i re c t c a l c u l a ti o n s h ow s t h a t t h e c o m m u t a t o r o f t w o L ie o p e r a t o r s , L f

Aa n d k g, is e q u a l t o t h e L i e o p e r a t o r a s s o c i a t e d w i t h t h e P o i s s o n b r a c k e t o f th eg e n e r a t i n g f u n c t i o n s g a n d f ; n a m e l y

A A A A

I L l , L , ] - - { . , { f , g } } - - k { : , , } - k { g , f } . (1 .3 .43)T h e c o m i n u t a t o r o f t w o L ie o p e r a t o r s is t h e r e f o r e a L i e o p e r a t o r a s w e ll. T h i s

Ais j u s t t h e b a s ic r e q u i s it e f or t h e l i n e a r s p a c e { k / } b e a L ie a l g e b r a u n d e rt h e c o m m u t a t o r p r o d u c t ( 1 .3 .4 2 ) . In a d d i t i o n , r e l a t i o n ( 1 .3 .4 3 ) s t a t e s t h a t t h ec o r r e s p o n d e n c e ( 1 .3 .4 0 ) is a n h o m o m o r p h i s m [9 .6 ], a s t h e a l g e b r a - g e n e r a t i n go p e r a t i o n b e t w e e n e l e m e n t s o f { f ( q , p ) } , i .e ., t he P o i s son b ra cke t { f , 9} , is i m -a g e d t o t h e a l g e b r a - g e n e r a t i n g o p e r a t i o n b e t w e e n t h e c o r r e s p o n d i n g e l e m e n t sA A Ao f { k f } , i .e ., t h e c o m m u t a t o r b r a c k e t [ k f , kg ].

A s a n e x a m p l e w i t h f = q2 ~ a n d h e n c e { f , g } - q p , we f ind :' 2 , g - - 2s - + + {s , W 11.4 Ha m ilton 's equations for the l ight-rayH a m i l t o n ' s e q u a t i o n s c a n b e c a s t in t e r m s o f P o i s s o n b r a c k e t s . I n f a c t, o naccoun t o f ( 1 .3 .30 ) , we m ay rewr i t e ( 1 .2 .13 ) i n t he su i t ab l e vec t o r fo rm

d u ( z ) - { u ( z ) H ( q , p , z ) } u ( z i ) - u i ( 1.4 1)d z ' '


20/58

2 0 L i n e a r R a y a n d W a v e O p t ic s n P h a s e S p a c e

t h e r a y -v e c t o r u ( z ) c o n t a i n i n g t h e z - d e p e n d e n t r a y v a r ia b l e s , w h o s e k n o w nv a l u e s ~ t t h e e n t r a n c e p l a n e a ,t z i a r e g r o u p e d i n t o th e v e c t o r u i .

T h e n , i n t r o d u c i n g t h e L i e o p e ra ,t or a s s o c i a t e d w i t h t h e o p t i c a l H a m i l t o n i a na c c o r d i n g t o ( 1 . 3 . 3 7 ) ; n a m e l y ,

-- ( O H O O H O )L , (z) - { . , H i - ~_~ Op~, Oq, Oq, , Op, , ' (1 .4 .2 )( )z -- :s 1,1 t

l ' ;(Is. (I ., I. I) at(: r ec as t i~ t() tl~(: (:()~(:is(: m~(l (:l(:ga.nt f()I'Iiid A ( z ) u ( z ) u ( z i ) u i ( 1 4 . 3 )d - ~ U ( Z ) - L . , -

wlli( :ll (:()lllt)int;S t)t)t, ll t, lt'~ vt;('t,()r a ll(l Lit'~-t)t)t;ra.t()r ~l ()ta ti()l l. F t)r xl()t,a,titmal( :(m( ' is t ;Iless , (rely t i le t ) ( ) ssit) lc ( tt ;I) (nl( t tnl( : ( ; ()f k , (hi z is ( 'xl) l i ( : i t ly s l l( )w n.As a,ll 'ea,tty rt;nm,rkt;( t, tll(; st)h ltit nl ()f E ft. (1 .4 .3) is (,ll( ' , m ls w er 1,() tll('~ flm (t a-ll le nt al t)I '()t)lt '~ni ()f gt; () me trit :al ()t)tit:s: 1,() (l( ;tel 'n line tile illt)ll(,-()ll(,t)ll(, rela ,ti( msf~n" a. giv(; ~ ()t)ti(: al ( lcvi(: c, wl~i(:l~ all() w ~ s t() fin(1 ()~ t tl~c lineal vc('t( )rs u,, f()rt l m t th e ra y trm ~ sfer rela,t i (n~s ui --~ uo ha ve s(m~(; ( l (~sire(l 1)r() t)er( ,ics.

F;( l lla. ti ( ) liS ()f t i le tyt )e (1 .4 .3) a, t ) l )ear i~ sev(;ra,1 a~(l ( l i t E 'ri~ g t i( : l ( ls ( )f l ) l~ysi( :s .A la.rge w tr ie ty ()f t)()tl~ a.~a.lyti(: a~ (l ~n ~( :ri(: a.1 n~el,l~()(ls ()f i~d,(:gra.ti(n~ lm.s |) ee n(levise(1, wh i( : h t)r()vi( l( ; exl) l i ( : i t ext)r( :ssi(n~s f()r exa( : t ( )r a l )I)r( )xi~ mt( ' a nalyti( "s()h ~ tions a.~(t ( t( '~fi~( ; gestu ral I)r()( : e(l~res f()r m~ mcr i( :a l a , lg()ri l ,ln~s witl~ ( te finite(le gre e ()f a( '( :~wa,(:y [~2, ~:t] . I~( lee(t , tl~( '~ i~ teg rati (n ~ ()f (1 .4 .3) w(n~l(l ( lese rve a,l)r() t ) er trea, t~ ( ;~ d, ( )f i ts ( )w~ . Ih 'r e w('~ will giv e a l i t t le i~sigld, i~t() t l~('~ l )r() lfle~n,will l )c l )~It ( )n t lm Ol)( :ra( , ( )r l l a , t l l l ' ( : ( )f (1 .4 .3) . T h e rea


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Ham iltonian Picture o f Light Optics. First-Ord er Ray Optics 21

t h e 4 - c o m p o n e n t r a y v e c to r u ( z ) e n t e r s ( 1 . 4 .3 ) i n p la c e o f t h e o n e f u n c t i o ny ( z ) a n d c o r r e s p o n d i n g l y t h e z - d e p e n d e n t L ie o p e r a t o r k , ( z ) in p la c e o f t h eo r d i n a r y z - f u n c t i o n a ( z ) . A s is w e l l k n o w n , o p e r a t o r s b e h a v e i n a s u b s t a n t i a l l yd i f f e r e n t w a y t h a n c - f u n c t i o n s , i . e . , f u n c t i o n s t h a t s e n d s c a l a r s t o s c a l a r s . O na p p r o a c h i n g t h e i n t e g r a t i o n o f E q . ( 1 .4 . 3 ) w e a r e u n a v o i d a b l y c o n f r o n t e d w i t ht h e o p e r a t o r n a t u r e o f kz , w h i c h i i la y al s o d e p e n d o n t h e a x i a l p a r a l n e t e r z .W h e n t h e L i e o p e r a t o r I_H is z - i n d e p e n d e n t , t h e s i m i l a r i ty w i t h t h e o r d i n a r yd i f f e r e n t i a l e q u a t i o n m a y b e b r o u g h t a s t e p f u r t h e r , s i n c e , a s w e w i l l s e e , t h es o l u t i o n t o ( 1 . 4 . 3 ) t a k e s a n e x p o n e n t i a l - l i k e f o r m .

I n f a ct , w h e n t h e H a m i l t o n i a n f u n c t i o n H d o e s n o t d e p e n d e x p l ic i t ly o nt h e a x i a l v a ri a b l e z , a s in t h e c a se o f p r o p a g a t i o n i n a h o m o g e n e o u s m e d i u mw h e r e n ( x , y , z ) - n o , t h e a s s o c i a t e d L i e o p e r a t o r I_H i s z - i n d e p e n d e n t a s w e l l .T h e s o u g h t s o l u t i o n c a n t h e n b e g i v e n t h e t r a n s p a r e n t e x p o n e n t i a l f o r mfu ( z ) - c (z -z ~ ) L - u i . ( 1 . 4 . 4 )T h e p r o p a g a t i o n o f t h e l i gh t r a y i s s e e n a s t h e e f f ec t o f t h e a c t i o n o f t h eio p e r a t o r c (Z -Z~) 'L , i , w h i c h i n p h a s e s p a c e t r a n s f e r s t h e p o i n t u i t o t h e p o i n tu ( z ) , a n d c o r r e s p o n d i n g l y i n r e a l s p a c e t h e r a y a t z i t o t h e r a y a t z .

S i n c e t h e e x p o n e n t i a l f u n c t i o n i s d e f i n e d f o r o p e r a t o r s e x a c t l y l i k e f o rs c a l a r s , t h e o p e r a t o r c ( z - z ~ ) L , is u n d e r s t o o d a s t h e p o w e r se r i es

~" ~ )JL J (1 .4 .5)( z - z i ) , k t i _ ( z - z i- j ! H "j= 0E x p o n e n t i a l s e ri e s o f L i e o p e r a t o r s , l ik e ( 1 .4 . 5 ) , a r e i n g e n e r a l k n o w n a s

L i e t r a n s f o r m a t i o n s [1 1]. T h e t h e o r y o f L i e t r a n s f o r m a t i o n s d e v e l o p s q u i t en a t u r a l l y f ro m t h e t h e o r y o f L ie o p e r a t o r s . I n f a c t, t h e L ie t r a n s f o r m a t i o n s{ c t L f ; t C IR} c o n s t i t u t e t h e L i e g r o u p a s s o c i a t e d w i t h t h e L i e a l g e b r a { L f } o ft h e L i e o p e r a t o r s ; r o u g h l y s p e a k i n g , w e m a y s a y t h a t t h i s g r o u p i s o b t a i n e db y e x p o n e n t i a t i n g t h e c o r r e s p o n d i n g L i e a l g e b r a t h r o u g h a r e a l p a r a m e t e r . I tis a c o m m o n l a n g u a g e i n t h e t h e o r y o f g r o u p s t o r e f e r to { L f } a s t h e a l g e b r ao f t h e g e n e r a t o r s o f t h e g r o u p o f t r a n s f o r m a t i o n s { c t L f ; t E R } . A d d i t i o n a lc o m m e n t s o n L ie t r a n s f o r m a t i o n s a r e g i v e n in w 2 .2 .2 .

I t is e a s y t o p r o v e t h a t t h e v e c t o r ( 1 . 4 . 4 ) is a s o l u t io n o f H a m i l t o n ' s e q u a -t i o n s ( 1 . 4 . 3 ) c o r r e s p o n d i n g t o t h e i n i t i a l v a l u e s u i . D i f f e r e n t i a t i n g b o t h s i d e so f ( 1 . 4 . 4 ) w i t h r e s p e c t t o z a n d s u m m i n g b a c k t o u ( z ) , w e e n d u p w i t h t h ed e s i r e d p r o o f

_ _ ( _ V ~ 1 7 6 z - z i ) J - l ' k j u i _ " L H 9 e ( z -z ~ ) ~ H u i - L u ( z ) ( 1 . 4 . 6 )d z U . Z . z ~ , ( j - I ) ' , , "j = l


22/58

2 2 L i n e ar R a y a n d W a v e O p t ic s n P h a s e S p a c e

H o w e v e r , t h e a b o v e v e c t o r p r o v i d e s o n l y t h e f o r m a l a n s w e r t o t h e p r o b -l e m o f r a y e v o l u t i o n , w h i c h a c t u a l l y i s s o l v e d o n c e t h e a c t i o n o f t h e o p e r a , to rc ( ~ - z ~ ) ' L - on the in i t ia ,1 v c ( : t o r u i i s e x p l i c i t l y d e f i n e d . T h e f u n c t i o n a , 1 f o r m o fc ( z - z ~ ) t . is o b v i o u s l y d e t e r m i n e d b y t h e H a m i l t o n i a n f l m c t io n H ( q , p ) , w h i c hin t u r n r c f lc ( ' t s tl~ c g e o m e t r y a ,n d o p t i c a l f c a , tu r c s o f t h e s y s t e m . I n o th e r w o r d s ,e x p r e s s io n ( 1 . 4 . 4 ) is t ) a s i ( 'a l l y a s e r i e s c x p a , n s io n , w h o s e t e r m s r e s u l t f r o m r e -t )ca, tc ( t ly P(fisscn~ 1)ra, ( :kcti~g H w it h u i . It, is th e t t am il to n ia n f l ln ( : t i ( )n th a t(l( '~t( 'rnfi**t's tl w ft)n** ()f tl w va,ri(n,s t( 'xn ,s i** tl,('~ se rie s m,( t lw**('(" th (' t)( )ss i|) ility()f S~ lml ni~ g l,l~(;~ 1,() a. (:l()s( ' (l ( 'xt)li(:it f() n~ .

A(l(lil,i(n~a,1 ( : ( ) l l l l l l ( ~ l l | ,S ( l( ;scl 'V ( ;S the ca ,s ( ; wh en t im t ta n f i l t on ia n ( lct ) (;n ( ls cx -I)li( :itly ()~ z, m~r s() a,ls( ) k~ (lo(;s. In tlm,t (:as r t,l~(; r( 'la,t,i()~ l)(;t,w('( '~ th e r a yvm' ial ) l t , s a t t l~ ( ; l ) lm~( ;s z i m~ (l z, a.s ( ;xI)I 't ;ss(;r i~ (1 .4 .4 ), is l l ( ) t l)r ()l) t ;r ly ( :or-r(;( ' t. Alt, l~()~gl~ m~ () I)(: rat() r-t ' t)rm r(;t)r(;s(;~d,a ti() n f()r t,l~(; (;v()hd, i(n~ ()f tim ra yva,ria,l)l( 's (:m~ stil l |)( ' g~u;ss(;(l, it ( 'm m () t |)('~ giv ('~ i~ t,l~(' f()rlll ()f t,t~(' si~ gl( ' L iet, ra.nsf()rnm,ti()l~ r i.(;., a.s t,h(; r (~f tl~c Li c () t)(; rat() r k , ov erth e wh () h; t)r()I)aga,ti(n~ l(;~gt, l~ fl ' (m~ z i t , ( ) z .

W( ' ~ a y i~ g ( ;~ ( ' ra l r ( ; la t ( ' t t~ ( ' ray wtI ' ia l f l( ' s a t z i m~(l z i~ t l~ ( ' ( ) l ) ( 'ra t(~r f () rn~Au ( , ) - ( , , ( 1 . 4 . 7 )

rcft c( :til lg l,ll~' vi~'w ~ f l, ll( '~ i~litial w; (' to r u i l~C illg l,ra,xlsfln'lll~;(l ild,~ u ( z ) t~y tll co tx ' ra . t ( ) r 9 J l ( z , z , i ) , wllir a,c(:~)r(tillgly is calh'(1 tll( ' r a y t 'ra 'n ,@ ', ' t" ope, ' t 'ato ' t"As t ) r( )ver Imf(~re, wlw ll t tw~ Ha ,m il toI l ian is z- in ( te l ) en( l ( ;n t , t t~c t ra .nsfer~I ) ( ; ra ,t ( ) r is ~ 'xa,~ : tly l 'e l) res~ ;~d ,e(t t~y th e Lic t r a , ns f~ r~ m ,t i~ (1 .4 .5 )"

A

@ ( z , - ( 1 . 4 . 8 )I f th e t ta l l l i l t~ n l im l ~ lel~ l lr (m Z, th is re t ) rcs cnta , t i (n l n~ longt' . r h~l( ts . T hi s

is du e to t, t lc fa , ( : t th a t t tw, a , r i th in et ic f ( )r o t )c ra , tors is not th e sa ,n le as fors (:a,la ,rs. ( ) I )er a ,t (~ r s a r e r th a t m ay ~ l ( )t ob ey the ( :o nmm ta , t ive law un de r

A A A A

mul t ip l i ( : a , t ion , a ,IM hen ( :c it, i s no t ge ne ra l ly t ru e tha , t AB - BA fo r opera , to r s ,w h i l s t i t i s a lw a y s t r im th a , t a b - b a for s ( : t f la , r s . Accordingly , ( ;x t ) ( )ncnt ia lo p e r a t o r s m a y n o t o b e y t h e s c m i g r o u p p r o p e r t y ; t h u s i t is n o t g en e r a l ly t r u et h a t c n . c s - - c n + B f o r o p e r a t o r s , w h i l s t i t is a l w a y s t r u e t h a t c ~ . c b - e ~ + b fo rs c a l a r s . T o g a , i n a f ee l ing fo r th i s p e c u l i a r a s p e c t o f t h e a r i t h m e t i c o f o p e r a t o r s ,w e a, r a n g c E q . ( 1 . 4 . 3 ) a s

Adu( ) - L ( 1 . 4 . 9 )S u b s t i t u t in g f o r d u ( z ) - u ( z + d z ) - u ( z ) , w e o b t a in

Au ( z + d z ) - [1 + L . ( z ) dz] u ( z ) - c L I - I z ) d z U ( Z ) , ( 1 . 4 . 1 0 )


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Ham iltonian Pictu re of L ight Optics. First-Order Ray Optics 23

w h e r e t h e e x p o n e n t i a l r e p r e s e n t a t i o n o n t h e r i g h t - h a n d s i d e i s i n a c c o r d w i t ht he power s e r i e s ( 1 . 4 .5 ) fo r i n f i n i t e s i m a l d z ' s . T h e a b o v e d e s c r i b e s t h e r a yp ropaga , t i on by t i l e i n f i n i t e s i m a l l eng t h f rom z t o z + d z . As no ted in w 1 .3.1 ,t h e p r o p a g a t i o n b y a f i n it e le n g t h c a n b e t h o u g h t o f a s a n i n f in i te c h a i n o fi n f i n i t e s i m a l p r o p a g a t i o n s l i k e ( 1 . 4 . 1 0 ) , s e q u e n t i a l l y o r d e r e d a c c o r d i n g t o t h eax i a l va r i ab l e . Thus , fo r i n s t ance , we have

A A AL ( z + d z ) d z ( z ) d zu [ ( z + d z ) + d z ] - c L H ( z N d z ) d z u ( z + d z ) - e , (?LH U(Z) . (1 .4 .11)

T h e t w o e x p o n e n t i a l o p e r a t o r s a b o v e m u l t i p l y t o o n e e x p o n e n t i a l o p e r a t o rc[C,(z+az)+ci i ( z )]az o n l y u n d e r t h e c o n d i t i o n t h a t t h e e x p o n e n t s L H ( z ) a n dk i~ ( z + dz ) c o i n m u t e w i t h e a c h o t h e r , i .e ., k/~ ( z ) k t ( z + d z ) - k ( z + d z ) k ( z ) .I n t h a t c a s e , s e q u e n t i a l l y c o n c a t e n a t i n g p r o p a g a t i o n s b y a l l t h e i n f i n i t e s i m a ls e g m e n t s d z ' s a l o n g t h e d e s i r e d d i s t a n c e f r o m z i t o z , w e e n d u p w i t h t h ee x p o n e n t i a l r e p r e s e n t a t i o n

(z , z i) - c "fz~ CI~ (z )az , (1 .4 .12)u a ( z ) d zi n a c e r t a i n a n a l o g y w i t h t i l e s o l u t i o n y ( z ) - - y ( z i) Y (Z i )CJ z~ of t i l e s ca l a r

h o m o g e n e o u s d i f f e r e n t i a l e q u a t i o n . T h i s i s n o t s u r p r i s i n g a s t h e c o m m u t a t i v i t ym akes ope ra t o r s c l o se r t o s ca l a r s . A l so , we r ega i n t he s i m p l e fo rm ( 1 .4 .8 ) s i nce ,f or a z - i n d e p e n d e n t g e n e r a t o r k , f z k /i d z - ( z - z i ) k u . E v i d e n tl y t h e r a yevo l u t i on i s found once t i l e ope ra t o r ( 1 . 4 .12 ) i s l e f t t o ac t o i l t i l e i npu t da t at o y i e l d t h e e x p l i c i t r a y - c o o r d i n a t e s a t z .

I n g e n e ra l , a s t h e L i e o p e r a t o r s k , ( z ) ' s a t d if f er e n t z d o n o t c o m m u t ew i t h e a c h o t h e r , t il e se x n ig r o up p r o p e r t y o f e x p o n e n t i a l s c a l a r s is n o t r e co v -e r e d w i t h i n t h e o p e r a t o r c o n t e x t . W e a r e t h e n c o n f r o n t e d w i t h t h e p r o b l e m o fs e e k i n g f o r a p r o p e r r e p r e s e n t a t i o n , e x a c t o r a p p r o x i m a t e , f o r t h e t r a n s f e r o p -e ra t o r 9 ) t ( z , z i ) , i. e ., w i t h t h e p ro b l em t o g i ve t he i n f i n i te p ro du c t o f z -o rd e redi n f i n i t e s i m a l t r a n s f o r m a t i o n s c c , ( z ) d z an exp l i c i t , pos s i b l y c l o sed , func t i ona lf o r m . R e a d e r s i n t e r e s t e d i n t h e d e t a i l e d a s p e c t s c o n c e r n i n g t h e i n t e g r a t i o n o fequ a t i on s o f t he t yp e ( 1 .4 .3 ) a re ad d res s ed t o [12, 13].

I n t h e f o r t h c o m i n g s e c t i o n s w e w i l l c o n s i d e r t h e r a y p r o p a g a t i o n t h r o u g hl inear ( o r f i r s t - o rde r ) o p t i c a l s y s t e m s , w h i c h c a n e f f e c t i v e l y b e d e s c r i b e d b yr a y - p a r a m e t e r i n d e p e n d e n t 4 4 m a t r i c e s . M o r e sp e c if ic a ll y, w e w i ll c o n s i d e rs y s t e m s , w h o s e s y m m e t r y p r o p e r t i e s a l l o w t o r e d u c e t h e d i m e n s i o n a l i t y o ft h e p r o b l e m f r om t h e 4 D t o t h e 2 D o p t ic a l p h a s e s p a c e, s o t h a t w e w i lln e e d o n l y a 2 x 2 m a t r i x t o d e s c r i b e t h e r a y p r o p a g a t i o n . W e w i l l s e e t h a tl i n e a r o p t i c a l s y s t e m s a r i s e f r o m H a m i l t o n i a n s t h a t a r e , o r a r e a p p r o x i m a t e da s , q u a d r a t i c p o l y n o m i a l s i n t h e p h a s e s p a c e v a r i a b l e s ; d e p a r t u r e s f r o m t h el i nea r i t y a re i den t i f i ed a s ab e r ra t ions [1 4]. T h u s , s e c o n d - d e g r e e p o l y n o m i a l s


24/58

2 4 L i n e a r R ay a n d W a v e O p t i c s n P h a s e S p a c e

a r e s e e n t o p la y a d i s t i n g u i s h e d r o le w i t h i n g e o m e t r i c a l o p t i cs ; i n d e e d , l i n e a ro p t i c a l s y s t e m s s t a n d a,s t h e c o n c r e t e r e a li z a ,t i on s o f H a m i l t o n i a , n d y n a m i c s ,g e n e r a t e( t | ) y p o l y n o m i a l s q u a d r a t i c i n t h e p h a s e s pa ,c e v a r i a b le s .

1.5 Lie transform ations in the opt ical phase spaceT h e ttm~fill ,~n>I,i~ ' f lWl~ mlis~ , { les(:ril){ '~l ix~ t, hc Im'viC nts se(:t i~ms, alh~ws ~ s t oi(l c~ tify a tm,tl~ t{~ ild,( :gra.t, (, I I m ~ i l t ~ ' s ~;~t~m.ti{n~s f in" tl~{ ', l ig ht ra y t{~ th e cor-rcs t) ol~ (l i~ g syl~ l)le( :l,i(: l, ral~sf~)r~m,t, iol~ i~ tl~( ; gt'~()~ cl, ri(:a,l-()t)l, ica,1 t)]m,sc st)a ,ce:

Exponent ia t ion ~ ( z' ' - zi ) L l l . ( 1 . 5 . 1 )I P o i s s o n b r ac k e t operation E l l . .. . ..

T h e r~'s~ll, il~g t)lm.s{ : st)a,c~, l,rm~sf{)r~na.l,i()~:A1 [ ] [ ( Z i ) b y ,, ( . . . . i )L // ' U ( Zo ) - ( ' ( : - z ' ) ' ~ l l U ( Z . i ) , ( 1 . 5 . 2 )

l " C f l ( ~ , ( : | , S [ , l l O , ew~llll,i(~ll ~ f tilt; ra y a l()llg l,ll( ', ~l)l,i(:al z- ax is .It, is (;vi~l~;ld, l,lm, , as excllq)lili( ',~l ill ( 1. 5. 1) fi)r ~t)l,i(:a,1 Ita,l~dlt,(~ nialls, w e

(: an g~',ll~',ral,(; 1)lms~'-Sl)aC~' l,ra,]lsf()rxl~ml,i()]ls, l m vi ll g ~ r ll~t, a (tir(~,('t, lillk w it h theZ-ev~)lllt, ~nl ~)f tl l~' ligllt, ray. We llmy ~:~)l~si~l~'x" m~y ~n~t, anu)usly ~liff~'rcntial)Ic,I)tias~;-slm.~:~' varia,l~l~; . [ ( q , p ) a,~ t t, lxe~ f~n '~ t ,l~e ~:~ n 'rest)~n~ling Lie l, rm> for~n a,-t, on, wl~ s( ' , " w di ( li ty " as a I)has(' ,-spa,(:(; l, ra,nsfi)r~m,l,i~)~ is (;sl,at)lisll( ;( l a(:( :or(ling1,() wl~(;l,l~er il, ( '~wre('l, ly ~m ,1)s l,l~(; g(; (~ el,ri( >( )I) l,it'a,1 l)t~a,s(;-st)a,('(; i~d,(~ its elf . W ere c al l i~ flu:t, tim,l, ti m ()l)tica.1 I)tm,se st)a,c e lm.s t,l~(', l)(W~flia.r 6 7 2 IR 2 st, ru( 'ta lre ,in C( )lll, asl , wil, l~ t,l~: IR4 str~ u:t ~ rc ~)f l ,l~e n~e(:ha,ni~:a,1 l) has c st)a( 'e.

W c wil l g iw; ,SOl~e cxm ~ t) l cs of Lic t ra ,x~sf (wl~m.t, i~n> in the o t) t, ical t )ha scst)a(:( ' ,, h~ l)a,rl,i(:~lar, i~ w 1.,5.1 an( t 1.5.2 w(; wil l (:t m sid cr a,s I)lm,sc-st)a.cefim(:ti~n~s . f ( q , p ) l ixma.r a n~ t qu ad ra t i c t )o l ynom iM s in q an d p . In tha . t case ,we cm~ ~ ;a .sily ~ ) |) ta il~ t l~c exp l i c i t exp ress ion s o f the t r a ,n s fo rm ed p lms c s pac eveer (w , s i~ l ) ly ~six~g l ,l~e ex t )onen t ia ,1 se r ies ex I )a ,n s io~ (1 .4 .5 ) . T he n in w 1 .5 .3we wi l l ] ) r ie l ty ~ les ( : r i l ) e the f ac to r iza t ion -ba , se ( t mcth{ )d wide ly u sed to dea lw i t h t h e L ie t r a n s fo r ~ n a t i o n s g e n e r a t e d b y h i g h e r - o r d e r p o l y n o m i a l s.

T o s i m p l i f y ~ o t a t io I ~ s w e w i ll c o n s i d e r t h e s i n g l e p a i r o f c o n j u g a t e v a r i a b l e s( q , p ) , a n d h e n c e w e w i ll a , c co u n t f o r t h e p r o p e r I - n , n ] x I R s t r u c t u r e o f t h er e l e v a n t 2 D p h a se - s pa . c e . W e w i ll d e n o t e b y ( t h e r e a l p a r a m e t e r o f t h e e x p o -..-..n e n t i a t i o n f r o m t h e L ie o p e r a t o r L , t o t h e L ie t r a n s f o r m a t i o n e C L f . W e w i l ln o t d e t a i l t h e p h y s i c a l m e a n i n g o f ~ , w hi ct~ d e p e n d s , o f c o u r s e , o n t h a t o fthe var ia ,b le f ( q , p ) i n o r d e r t h a t t h e e x p o n e n t , ~ k , b e d i m e n s i o n l e ss . T h u s , i ff ( q , p ) c a n b e i n t e r p r e t e d a s a n o p t i c a l H a m i l t o n i a n , ~ m a y a c q u i r e t h e m e a n -i n g a ,s t h e e v o l u t i o n v a r i a b l e , a n d a c c o r d i n g l y t h e r e l e v a n t L i e t r a n s f o r m a t i o nt h a t K s t, h e r a y - t r a n s f e r o p e r a t o r .


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H a m i l t on i a n P i c t u r e o f L i g ht O p t ic s . F i r s t- O r de r Ra y O p t i c s 2 5

q T(" ; :: ................t , , , , I . , " i p . "-

(a) (b)r

Z

F I G U R E 1 .5 . A s t h e t i p o f t h e r a y - m o m e n t u m v e c t o r n = ( p, p z ) i s c o n f i n e d o n t h e c ir c le o fr a d i u s n , ( a ) t h e " m o m e n t u m t r a n s l a t i o n " is a r o t a t i o n o f t h e v e c t or , w i t h t h e t i p s l i p pi n ga l o n g t h e c i r c l e , w h i c h a m o u n t s t o ( b ) a v a r i a t i o n o f t h e d i r e c t i o n o f t h e r a y w i t h r e s p e c t t ot h e q -z r e f e r e n c e a x e s .

1 . 5 . 1 L i n e a r p o l y n o m i a l s i n q a n d pT h e m o m e n t u m v a r i a b l e p g e n e r a t e s a s hi ft o f t h e p o s i t i o n c o o r d i n a t e o f t h ep h a s e - s p a c e v e c t o r , w i t h o u t a f f e c t i n g t h e r e l a t i v e m o m e n t u n i c o o r d i n a t e . I nfac t

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